unfolding a tangent equilibrium-to-periodic heteroclinic...

SIAM J. APPLIED DYNAMICAL SYSTEMS c© 2009 Society for Industrial and Applied MathematicsVol. 8, No. 3, pp. 1261–1304

Unfolding a Tangent Equilibrium-to-Periodic Heteroclinic Cycle∗

Alan R. Champneys†, Vivien Kirk‡, Edgar Knobloch§, Bart E. Oldeman¶, and

J. D. M. Rademacher‖

Abstract. The dynamics occurring near a heteroclinic cycle between a hyperbolic equilibrium and a hyperbolicperiodic orbit is analyzed. The case of interest is when the equilibrium has a one-dimensionalunstable manifold and a two-dimensional stable manifold while the stable and unstable manifoldsof the periodic orbit are both two-dimensional. A codimension-two heteroclinic cycle occurs whenthere are two codimension-one heteroclinic connections, with the connection from the periodic orbitto the equilibrium corresponding to a tangency between the two relevant manifolds. The resultsare restricted to R

3, the lowest possible dimension in which such a heteroclinic cycle can occur,but are expected to be applicable to systems of higher dimensions as well. A geometric analysis isused to partially unfold the dynamics near such a heteroclinic cycle by constructing a leading-orderexpression for the Poincare map in a full neighborhood of the cycle in both phase and parameterspace. Curves of orbits homoclinic to the equilibrium are located in a generic parameter plane, asare curves of homoclinic tangencies to the periodic orbit. Moreover, it is shown how curves of folds ofperiodic orbits, which have different asymptotics near the homoclinic bifurcation of the equilibriumand the homoclinic bifurcation of the periodic orbit, are glued together near the codimension-two point. A simple global assumption is made about the existence of a pair of codimension-twoheteroclinic cycles corresponding to a first and last tangency of the stable manifold of the equilibriumand the unstable manifold of the periodic orbit. Under this assumption, it is shown how the locus ofhomoclinic orbits to the equilibrium should oscillate in the parameter space, a phenomenon knownas homoclinic snaking. Finally, we present several numerical examples of systems that arise inapplications, which corroborate and illustrate our theory.

Key words. global bifurcation, Shil’nikov analysis, heteroclinic cycle, homoclinic orbit, homoclinic tangency,snaking

AMS subject classifications. 34C23, 34C37, 37C29, 37G20

DOI. 10.1137/080734923

∗Received by the editors September 9, 2008; accepted for publication (in revised form) by H. Kokubu July 9,2009; published electronically September 23, 2009. This work was supported in part by the New Zealand Institute ofMathematics and Its Applications, the Weierstrass Institute Berlin, the University of Auckland Research Committee,the UK Engineering and Physical Sciences Research Council, and the National Science Foundation grant DMS-0605238.

http://www.siam.org/journals/siads/8-3/73492.html†Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, United Kingdom (a.r.

[email protected]).‡Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand (kirk@math.

auckland.ac.nz).§Department of Physics, University of California, Berkeley, CA 94720 ([email protected]).¶Computer Science and Software Engineering, Concordia University, 1455 de Maisonneuve Blvd. W., Montreal,

QC, H3G 1M8, Canada ([email protected]).‖Department MAS, Centre for Mathematics and Computer Science (CWI), Science Park 123, 1098 XG Amster-

dam, The Netherlands ([email protected]). This author acknowledges support from a PIMS fellowship, theSPP 1095 program of the German Research Foundation DFG, and the NDNS+ program of the Dutch ResearchFoundation NWO.

1261

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1262 CHAMPNEYS, KIRK, KNOBLOCH, OLDEMAN, AND RADEMACHER

1. Introduction. We are interested in the dynamics near a heteroclinic cycle connectinga hyperbolic equilibrium solution and a hyperbolic periodic orbit, referred to here as an EP-cycle. Specifically, we consider dynamics in R

3 and assume that the equilibrium (denoted E)has a one-dimensional unstable manifold W u(E), while the periodic orbit (denoted P ) hasa two-dimensional unstable manifold W u(P ). In this case, a heteroclinic connection from Eto P generically will be of codimension one, while a connection from P to E generically willbe of codimension zero. We denote by EP1-cycle an EP-cycle in R

3 where the connectionsfrom E to P and from P to E are both of the appropriate generic type. Such a heterocliniccycle as a whole is then a codimension-one object; i.e., it occurs at isolated points in a genericone-parameter family of vector fields. In a two-parameter setting, EP1-cycles will occur onone-dimensional curves in the two-parameter space, with one possible endpoint of a curve ofEP1-cycles occurring when a tangency develops between W u(P ) and W s(E); we shall refer tosuch a codimension-two cycle as an EP1t-cycle and call the point in a two-parameter spaceat which an EP1t-cycle exists an EP1t-point.

EP1t-points arise in a number of applications. For instance, Champneys et al. [4] studyways in which branches of isolated traveling pulses interact with small-amplitude waves bornin a Hopf bifurcation, in the context of excitable systems. By passing to a moving frameof reference, they reduce this question to the study of various codimension-two mechanismsfor the termination of curves of homoclinic bifurcations as a Hopf bifurcation is approached.In two of the models discussed in that paper, i.e., the FitzHugh–Nagumo equations and anine-dimensional model of intracellular calcium dynamics, it is found that an EP1t-point actsas an organizing center for homoclinic bifurcations of equilibria. In the parameter plane of theFitzHugh–Nagumo model many different branches of homoclinic orbits appear to have singlefolds near an EP1t-point, while in the calcium model a single branch of homoclinic orbits foldsmany times (in a “homoclinic snake”) near a pair of EP1t-points. EP1t-points have also beenseen in a three-dimensional model motivated by studies of semiconductor lasers with opticalreinjection [20]. In this case, folds of a single curve of homoclinic orbits again accumulateon a pair of EP1t-points in a manner similar to that seen in the nine-dimensional calciummodel in [4]. The theoretical study of EP1t-points presented here is motivated by our desireto explain the observed folding and accumulation of homoclinic bifurcation curves in theseand other systems.

Our main aim in this paper is to provide a geometric analysis that unfolds the dynamicsnear an EP1t-cycle. We are interested in how curves of codimension-one homoclinic bifurca-tions in a suitable parameter plane emanate from an EP1t-point. Specifically, we locate curvesof homoclinic tangencies to the periodic orbit (which we call P -homoclinic tangencies, withnontangent homoclinic orbits being called P -homoclinics) and homoclinic orbits to the equi-librium (E-homoclinics). We also investigate bifurcations of periodic orbits near EP1t-points.Curves of folds of periodic orbits in a parameter plane are known to exist with different asymp-totics near P -homoclinic tangencies and E-homoclinics (see, for example, [13, 10]); we explainhow such curves are glued together near an EP1t-point. Motivated by several examples ofsnaking curves of E-homoclinic orbits observed in applications, we explain geometrically howsuch behavior can arise when there is a curve (in two-parameter space) of EP1-cycles, whoseextrema are EP1t-points. Such a geometry would occur when the two-dimensional manifoldsW u(P ) and W s(E) are “pulled through” each other as we vary a second parameter. Finally,

UNFOLDING A TANGENT E-TO-P HETEROCLINIC CYCLE 1263

we present several numerical examples of systems that arise in the applications discussed inthe previous paragraph which corroborate and illustrate our theory.

Our method of analysis follows a standard procedure. We first construct a model of thedynamics near an EP1t-cycle in terms of Poincare maps between neighborhoods of E and P ,and then analyze the ensuing algebraic bifurcation equations. Near each of E and P we derivelocal maps by assuming that the flow is governed by appropriate linearized vector fields inthese regions. The flow near the heteroclinic connections between E and P is approximatedby linearizing about each heteroclinic connection, which enables us to derive two global maps.Appropriate composition of the local and global maps yields the desired return maps. While wedo not give complete proofs that this model approximates the dynamics near the heterocliniccycle, we believe a rigorous proof is possible, for instance, by way of the methods describedin [15, 29, 30]. Our aim instead is to provide a direct and intuitive geometric picture andto uncover the scaling laws of the separate codimension-one bifurcations that occur near thecodimension-two point.

In this paper we restrict our attention to the lowest possible phase space dimension,namely, R

3, capable of supporting an EP1t-point. However, the results we obtain are likely toapply to systems in higher dimensions where the equilibrium and periodic orbit have additionalstrongly stable directions. Results such as the so-called homoclinic center manifold theorem[22, 32, 37] or Lin’s method for EP-cycles [28, 29, 30] may well provide the required extensionto higher dimensions.

There are a number of previous theoretical studies on EP-cycles (also called singularcycles) from an ergodic theory point of view, focusing on properties of nonwandering sets (see[1, 23, 24, 26] and references therein). Such properties were studied for EP1t-cycles in [25]when the eigenvalues at E are real with a stable leading eigenvalue, and the Floquet multipliersat P are positive. Our results for this case are complementary, providing details on parametercurves for certain specific solutions, as described in sections 2 and 3. Moreover, we also studythe case where the leading eigenvalue at E is unstable, as well as complex eigenvalues at Eand negative Floquet multipliers at P .

The outline of the paper is as follows. Section 2 contains a summary of our main findingsabout the bifurcations seen near an EP1t-point. Details about construction of the returnmaps and results on the organization of P - and E-homoclinic orbits near an EP1t-point arecontained in sections 3–5. Section 6 contains results on the dynamics of the complete returnmap near an EP1t-cycle, specifically on the locations of folds of periodic orbits. Section 7considers the more global analysis which motivates the snaking bifurcation diagrams. Section 8contains numerical results on example systems, and, finally, section 9 draws conclusions andsuggests avenues for future work.

2. Approach and main results. We use a standard method of analysis to unfold an EP1t-point. That is, we define appropriate Poincare sections close to E and P in order to reduce thedynamics nearby (in phase and parameter space) to the composition of various local and globalmaps, as shown in Figure 1. Once the maps are constructed, we use them to determine thelocation in parameter space of single-pulse E-homoclinic orbits and P -homoclinic tangencies,as well as folds of primary periodic orbits. By “single-pulse” we mean E-homoclinics that donot come close to E other than as t → ±∞, although they may visit arbitrarily close to P ;single-pulse P -homoclinics visit a neighborhood of P only as t → ±∞. By “primary periodic


orbits” we mean those that form a sequence converging to a single-pulse E-homoclinic orP -homoclinic tangency. On the level of these maps our analytical results are rigorous exceptfor the loci of folds of periodic orbits for complex eigenvalues at E.

Since locating single-pulse E-homoclinics involves tracking only the unstable manifold ofE, the dynamics in a neighborhood of E will not affect the loci of E-homoclinics in parameterspace. Similarly, the dynamics near P will not affect the parametric curves of P -homoclinictangencies. Thus, for the purposes of finding single-pulse E-homoclinics and P -homoclinictangencies we need to consider only three cases. Case I, where the eigenvalues of E are realand the Floquet multipliers of P are positive, is discussed in section 3. Case II, with realeigenvalues of E and negative Floquet multipliers of P , follows in section 4. Case III, withcomplex eigenvalues of E and positive Floquet multipliers of P , forms the subject of section 5.Results for the case where E has complex eigenvalues and P has negative Floquet multipliersfollows without further work, since the location of E-homoclinics will be as in Case II and thelocation of P -homoclinic tangencies will be as in Case III.

We now summarize our main findings on the leading-order geometry and scaling laws ofbifurcation sets; assumptions, derivation, and details of these results are contained in sections3–7. We work in a two-parameter space, with parameters α and β. We assume that anEP1t-cycle exists when α = 0 and β = 0, and that α unfolds the codimension-one heteroclinicconnection from E to P while β independently unfolds the heteroclinic tangency from P toE, as in Figure 4. Thus, codimension-one connections from E to P occur on the β-axis,tangencies between W u(P ) and W s(E) occur on the α-axis, and EP1-cycles occur on eitherthe positive or negative β-axis. We use a constant k (if the eigenvalues of E are real) or k (ifE has a pair of complex eigenvalues) to indicate on which half of the β-axis the EP1-cyclesoccur; we define k or k to be positive if transverse heteroclinic connections from P to E occurfor β < 0.

We adopt the following notation: for E, λ and μ are the absolute values of the real partsof the leading stable and unstable eigenvalues, respectively; for P , the period of the orbit isT = 2π/Ω and the Floquet multipliers are L = se−lT and M = semT , where s = ±1 and l,mare real and positive. Moreover, δE ≡ λ/μ, δP ≡ l/m while k denotes either k or k, dependingon the case. The case of complex multipliers at P cannot occur in R

3 if P is of saddle type.All of the following results are for α and β small.

E-homoclinic orbits (Figures 5 and 10). E-homoclinic orbits occur at

α = α±n (β) = αn

(1 ± m

Ω

√βn − β

k

),

to lowest order, where αn = KM−n, βn = SKδP Ln for constants K and S, and n is the (ap-proximate) number of complete circuits that the E-homoclinic orbit makes in a neighborhoodof P .

Thus E-homoclinic orbits lie on a countable family of parabolas, with the different pa-rabolas distinguished by the number of circuits n that the corresponding E-homoclinic orbitsmake about P . The family of parabolas accumulates on the half-line of EP1-cycles. To lead-ing order as n → ∞, the tips of the parabolas occur at α = KM−n and lie on the curveβ = sgn(α)S|α|l/m. All the parabolas have the same orientation, with vertices pointing either


up or down, depending on the sign of k. Different cases can be distinguished, depending onthe Floquet multipliers at P . In R

3, since the flow around P must be orientation-preserving,and P is of saddle type, there are only two generic possibilities:

1. Positive Floquet multipliers at P : Here all parabola tips occur for the same sign of α.See Figure 5.

2. Negative Floquet multipliers at P : The sign of α at which parabola tips occur alternatesas n is incremented. See Figure 10.

P -homoclinic tangencies (Figures 8, 12, 15, and 16). There are two cases, dependingon the eigenvalues at the equilibrium.

1. Real eigenvalues at E: P -homoclinic tangencies occur for

α = −cβδE , β > 0,

to lowest order, where c is a constant; P -homoclinic orbits exist in the region boundedby this curve and the half-lines α = 0, β < 0 (if k > 0) or α = 0, β > 0 (if k < 0)where EP1-cycles exist, and so are present on both sides of the α-axis only if k > 0.See Figure 8.

2. Complex eigenvalues at E: P -homoclinic tangencies come (asymptotically) in twotypes, which we refer to as tip- and side-tangencies. Tip-tangencies lie on a “snaking”curve

α = −cβδE cos(

Φ − ω

μln β

), β > 0,

where c and Φ are constants. At the folds of this curve the asymptotics are notvalid. Side-tangencies occur on near-vertical curves, a pair of which emanates fromthe vicinity of each of the folds of the tip-tangency curves, oriented in the positive β-direction for k < 0 and in the negative direction for k > 0. P -homoclinic orbits exist atall points in the region bounded by the curves of tangencies, with many P -homoclinicscoexisting in some regions. We refer the reader to section 5 for details. The overallpicture is like the laying down of a long folded cloth, viewed from an oblique angle,with the fold length becoming zero as we approach the floor. See Figures 12, 15,and 16.

Fold bifurcations (Figures 19 and 20). We consider only positive Floquet multipliers atP . It is well known that there must be infinitely many fold bifurcations (and also period-doubling bifurcations) at parameter values near a P -homoclinic tangency. On the other hand,in the case of complex eigenvalues at E with δE < 1, there are infinitely many folds (andperiod-doublings) that accumulate on each E-homoclinic orbit. These two families of foldsmust somehow glue together at an EP1t-point. How this gluing occurs gives insight intoa complete unfolding of the dynamics near the codimension-two point. There are severalcases depending on the eigenvalues at E. The results for complex eigenvalues come from asimplification that we cannot rigorously justify in general but nevertheless match the numericalresults well in all cases.

1. Real eigenvalues at E with δE < 1. Here each parabolic curve of E-homoclinic orbitshas an inclination flip bifurcation near its tip where the twistedness of the global


stable manifold of E around the E-homoclinic orbit changes. Such codimension-twobifurcations are each the root point of a curve of folds that is associated with theP -homoclinic tangency. See Figure 19(a),(b).

2. Real eigenvalues at E with δE > 1. This case is similar in that there is one curve offolds associated with each E-homoclinic parabola, but this fold remains close to onebranch of the parabola rather than terminating at an inclination flip point. See Figure19(c),(d).

3. Complex eigenvalues at E with δE > 1. This case is similar to case 2, although theremay now be finitely many additional curves of folds associated with each E-homoclinicparabola, which disappear in pairs at cusp points. See Figure 20(a).

4. Complex eigenvalues at E with δE < 1. Here there are infinitely many curves of foldsassociated with each E-homoclinic parabola. The folds disappear in pairs at cusppoints. See Figure 20(b)–(d).

Homoclinic snaking (Figures 21–25). We give a geometric explanation of the homoclinicsnaking phenomenon that has been observed in a number of examples. Our explanation isbased on certain hypotheses about the properties of the heteroclinic connection from P toE. Specifically, we assume that as the parameter β varies there is a first and a last tangencybetween W u(P ) and W s(E). The role of α is as before. Assuming a connection from Eto P exists, transverse intersections between W u(P ) and W s(E) will give rise to EP1-cyclesand occur in pairs on an interval of the β-axis, while tangencies between W u(P ) and W s(E)will correspond to EP1t-cycles and will occur at isolated points on the β-axis, including theendpoints of the interval of EP1-cycles. From this geometry we have two cases:

1. Positive Floquet multipliers at P : There is a single smooth curve of E-homoclinicorbits having one sign of α that accumulates on the curve of EP1-cycles. The curveof E-homoclinics has infinitely many extrema with respect to β, which converge onthe EP1t-points in the manner described above in the local analysis. See Figures 23and 24. Such a curve has been called a homoclinic snake.

2. Negative Floquet multipliers at P : There is one curve of E-homoclinic orbits in the(α, β)-plane for each sign of α. Each curve has infinitely many extrema with respectto β accumulating on each EP1t-point. See Figure 25.

3. Case I: Real eigenvalues at E, positive Floquet multipliers at P . The assumptionswe make in this case are as follows. The heteroclinic connections lie in the leading stable andunstable directions at E and P , where applicable, and the tangency of the stable manifold ofE with the unstable manifold of P is quadratic. The Jacobian matrix of the flow evaluatedat E has real distinct eigenvalues μ, −λ1, and −λ2 satisfying −λ2 < −λ1 < 0 < μ and thenonresonance conditions

m1λ1 + m2λ2 − m3μ /∈ {−μ, λ1, λ2}

for all m1,m2,m3 ∈ N satisfying m1 + m2 + m3 > 1.For the Floquet multipliers exp(−lT ) and exp(mT ) at P , where T is the period of P , the

distinct rates l and m are (real) positive constants satisfying

m1m − m2l /∈ {m,−l}


(a)

PΣ0

W u(P )

E

Σ1

Σ3

Σ2

(b)

Σ0

Σ1

Σ3

Π3

Σ2

Π1

Π2Π0

Figure 1. Case I: (a) Sketch of phase space and sections at E and P ; the situation for k < 0 is shown.(b) Decomposition into local and global maps.

y

ψ

ξ

ηx

z

PE

h2

h3

Ch1

h0

Figure 2. Schematic diagram showing the local coordinate systems used near E and P .

for all m1,m2 ∈ N with m1 + m2 > 1.Finally, the underlying vector field is CN smooth with N large enough for the results

in [36] to apply near E and P with a C1 conjugacy. See (3.1) and (3.3).

3.1. Map construction. We first construct local maps near E and P . Following Sternberg[36], the nonresonance assumption allows us to choose local coordinates (x, y, z) near E withthe coordinate axes pointing in the directions of the associated eigenvectors so that the flownear E can be written as

(3.1) x = −λ1x, y = −λ2y, z = μz.

Cross-sections near E are defined using the local coordinates, i.e.,

Σ0 = {(x, y, z) | x = h0} and Σ1 = {(x, y, z) | z = h1},

where h0 and h1 are small positive constants. See Figure 2. For sufficiently small h0 andh1 the flow induces a map from Σ0 to Σ1. An approximation to the map is obtained byintegrating (3.1). We obtain

(3.2) Π0(h0, y, z) =

(h0

(z

h1

)δE

, y

(z

h1

)δE

, h1

), z > 0,

whereδE = λ1/μ


measures the relative importance of the leading stable eigenvalue of E compared to its unstableeigenvalue, and

δE = λ2/μ > δE .

We construct a local map near P in a similar way. To define local coordinates near P bysmooth conjugacy, we note first that P can be parametrized by an angle ψ with −π < ψ ≤ π(we pick an origin for the ψ-coordinate below). Next, we define a cross-section C transverseto P at ψ = 0. Then W u(P ) and W s(P ) will intersect C on one-dimensional curves. Wechoose local coordinates (η, ξ) in C so that W u(P ) ∩ C and W s(P ) ∩ C are tangent to thecoordinate axes, with the origin of each coordinate at P ∩ C. See Figure 2. We can extendthese local coordinates to a neighborhood of the whole of P by evolution under the flow in anobvious way. Our aim is to find coordinates in which the flow near P takes the form

(3.3) ψ = Ω, η = mη, ξ = −lξ,

where Ω ≡ 2π/T with T the period of P .To do so, we choose coordinates that trivialize the ψ dynamics by straightening stable

and unstable fibers (see, e.g., Lemma 1 in [2] for a symmetric version of this procedure) andby rescaling time with a suitable Euler multiplier. In the resulting planar system we choosecoordinates that linearize the hyperbolic period-T map; the nonresonance assumption on land m guarantees that this is possible. See, e.g., [36]. Finally, we note that this period-T mapis trivially generated by (3.3), and we can pull back the coordinate change to the full space.

We define further cross-sections near P using the local coordinates

Σ2 = {(ψ, η, ξ) | ξ = h2} and Σ3 = {(ψ, η, ξ) | η = h3},

where h2 and h3 are small positive constants (we will constrain the constant h2 further, asdiscussed below). For sufficiently small h2 and h3, the flow near P induces a map from Σ2 toΣ3, with an approximation to the map being obtained by integrating (3.3). To lowest order,we find

(3.4) Π2(ψ, η, h2) =

([ψ − Ω

mln(

η

h3

)]mod 2π , h3 , h2

(η

h3

)δP)

, η > 0,

whereδP = l/m

measures the relative importance of the stable Floquet exponent compared with the unstableexponent.

We next construct the global maps. First, we define a map Π1 : Σ1 → Σ2 that approx-imates the dynamics near the heteroclinic connection from E to P . The connection, whenit occurs, is one-dimensional, intersecting Σ1 at (x, y, z) = (0, 0, h1) and intersecting Σ2 at(ψ, η, ξ) = (ψ, 0, h2), where ψ is a real constant. The connection is of codimension one, andwe introduce a real parameter α that unfolds the heteroclinic bifurcation, which we assumeto occur at α = 0. Retaining just the lowest order terms in x, y, and α, in the standard way,we obtain the map

(3.5) Π1(x, y, h1) = (ψ + ax + by + fα , cx + dy + α , h2),


W u(P )

β < 0

z

W s(E)

z

W s(E)

z

W s(E)

k > 0

k < 0

z z

W s(E)y W s(E)W s(E)

W u(P )

W u(P )W u(P )

β = 0

z

β > 0

Σ0

y

Σ0

y y

Σ0

y

Σ0 Σ0

y

Σ0

y

W u(P )W u(P )

Figure 3. Case I: Unfolding a heteroclinic tangency between W u(P ) and W s(E). Each panel shows therelative positions of W u(P ) ∩ Σ0 and W s(E) ∩ Σ0. Upper row: k > 0. Lower row: k < 0. Left column: β > 0.Middle: β = 0. Right: β < 0. The large dots correspond to heteroclinic orbits from P to E.

where a, b, c, d, f are generically nonzero constants with ad−bc �= 0, and where we have chosena scaling so that α has a coefficient 1 in the expression cx + dy + α. As we will see later, weare free to set ψ = 0.

We now derive a map Π3 : Σ3 → Σ0 that approximates the dynamics near a heteroclinictangency between W u(P ) and W s(E). The tangency generically will be of codimension one,and so we introduce a real parameter β to unfold the heteroclinic tangency, which we assumeto occur at β = 0, as illustrated in Figure 3.

So far we have not specified an origin for the ψ-coordinate near P . We now pick the originof ψ in Σ3 to correspond to the heteroclinic tangency between W u(P ) and W s(E). Then,when β = 0, the heteroclinic tangency intersects Σ3 at (ψ, η, ξ) = (0, h3, 0) and intersects Σ0

at (x, y, z) = (h0, y, 0) for some constant y. Since ψ parametrizes W u(P ) ∩ Σ3 and since thetangency between W u(P ) and W s(E) is generically quadratic, the y- and z-coordinates ofW u(P ) ∩ Σ0 near the tangency can be written as

y = y + gψ, z = kψ2,

to lowest order in ψ, where g and k are constants with k �= 0. Note that ψ is small near thetangency, although not small over a full neighborhood of P . In the standard way, we extendto orbits near the heteroclinic tangency and express the map Π3 to lowest order in ψ, ξ, andβ as follows:

(3.6) Π3(ψ, h3, ξ) = (h0 , y + gψ + qξ + rβ , kψ2 + sξ + β + vβψ),

where q, r, s, v are constants with q − rs �= 0, and where we have chosen a scaling so that βhas coefficient 1 in the expression kψ2 + sξ + β + vβψ.

Having fixed the origin of the ψ coordinate near P , we now pick h2, the constant deter-mining the surface Σ2, to simplify the geometry and maps further. Specifically, we pick h2


tangency

P →E

β

EP1tP →E

E→P

α

P →E

EP1

Figure 4. Locations of heteroclinic connections, cycles, and tangencies in the (α, β)-plane for k > 0 in thecase that E has real eigenvalues. When E has complex eigenvalues, this sketch corresponds to the case k > 0,as discussed in section 5.

so that ψ = 0, i.e., so that when the heteroclinic connection from E to P occurs, the phaseof the heteroclinic connection is zero at the point where it crosses Σ2. This ensures that thecross-sections Σ2 and Σ3 are centered about the same ψ value, which greatly simplifies theanalysis near P , particularly in the case of negative Floquet multipliers of P , as discussedin section 4. We note that it is possible to satisfy this constraint on h2 while picking h2

arbitrarily small.Figure 4 summarizes the consequences of the assumptions built into our map construc-

tion for the location in the (α, β)-plane of the various heteroclinic connections and cycles.Consider the case k > 0 (the case k < 0 is obtained simply by reflection about the α-axis). The codimension-two EP1t-cycle occurs at the origin. For every point in the negativeβ half-plane, there exists at least one heteroclinic connection from P to E. These are ro-bust as codimension-zero objects and become heteroclinic tangencies when β is zero. Thecodimension-one heteroclinic connections from E to P occur on the β-axis, and so EP1-cyclesoccur on the negative β-axis.

3.2. E-homoclinic bifurcations. We use the maps derived above to locate E-homoclinicsnear the EP1t-point. In terms of local coordinates near E, an E-homoclinic first crosses Σ1

at (0, 0, h1) and returns to Σ0 with z-coordinate equal to zero. Thus we find E-homoclinicsby equating to zero the z-component of Π3 ◦ Π2 ◦ Π1(0, 0, h1). We find that the followingapproximate equation must be satisfied if an E-homoclinic is to exist:

k

[(Ψ + fα − Ω

mln α

)mod 2π

]2

− SαδP + β

+ vβ

[(Ψ + fα − Ω

mlnα

)mod2π

]= 0,(3.7)

where S = −sh2h−δP3 and Ψ = (Ω/m) ln h3. Solutions of this equation are given by

(3.8)(

Ψ + fα − Ωm

ln α

)mod2π = −vβ

2k± 1

2

√v2β2

k2+

4k(SαδP − β).

We expand (3.8) to obtain, to lowest order,

(3.9) Ψ − Ωm

ln α = 2nπ − vβ

2k±√

S

kαδP − β

k,


α

β

α

β

α

β

α

βS > 0k < 0

S < 0k < 0

S > 0k > 0

S < 0k > 0

Figure 5. Case I: E-homoclinic orbits occur locally on the solid curves. All panels show the case δP > 1;the case δP < 1 differs only in the shape of the dashed curve β = SαδP , which is the envelope of the folds ofE-homoclinic loci.

where n is an integer counting the number of times the homoclinic orbit winds around P .There are double roots on the right-hand side of (3.9) when β = SαδP . Thus, when k > 0,E-homoclinics exist only for β < SαδP , to leading order, and annihilate pairwise on the curveβ = SαδP , α > 0. Rearranging (3.9), we find that the annihilation points occur, to leadingorder, at

α = αn = KM−n, β = βn = SKδP Ln,

where K = exp(Ψm/Ω), M = exp(2mπ/Ω), and L = exp(−2lπ/Ω). In these expressions Mand L are, respectively, the unstable and stable Floquet multipliers of P . At α = αn, twosolutions emerge for varying β for approximately the same value of α. Specifically, substitutingβ ≈ βn and expanding, we find curves

α±n (β) = αn

(1 ± m

Ω

√βn − β

k

)so that the sign of β−βn is opposite that of k. In fact, there are eight distinct cases dependingon the signs of δP − 1, S, and k. See Figure 5.

These conclusions are in accord with the results on nonwandering sets in [25]: the regionin which E-homoclinic orbits exist, bounded by the envelope curve and the β-axis, lies in thecomplement of the region for which [25] finds the nonwandering set to be trivial (compareFigure 8 for S > 0, k > 0 with Figure 2 in [25]).

3.3. P -homoclinic bifurcations. In terms of local coordinates near P , a P -homoclinicorbit first crosses Σ3 with ξ = 0 and returns to Σ2 with η = 0. Thus, P -homoclinics are foundby equating to zero the η-component of Π1 ◦ Π0 ◦ Π3(ψ, h3, 0):

(3.10) η = α + cZδE + d(y + gψ + rβ)Z δE = 0,


W u(P )

ψ

W u(P )

W s(P ) W s(P )

W u(P )

β > ββ < β

W s(P ) W s(P )

W u(P )W u(P )

c > 0

W s(P )y

W u(P )

β = β

η

c < 0

k > 0

k > 0

η

η

η η

η

ψ ψ

ψψψ

Σ2 Σ2

Σ2 Σ2 Σ2

Σ2

W s(P )

Figure 6. Case I: Unfolding a homoclinic tangency between W u(P ) and W s(P ) for fixed, nonzero α. Eachpanel shows the relative positions of W u(P ) ∩ Σ2 and W s(P ) ∩ Σ2. At tangency, β = β(α) ≡ (−α/c)1/δE ,to lowest order. The upper row has c > 0, k > 0, while the lower row has c < 0, k > 0. The left column isfor β < β, the middle column for β = β, and the right one for β > β. The large dots correspond to orbitshomoclinic to P . The cases c > 0, k < 0 (c > 0, k < 0) locally look like the lower (upper) row but with the leftand right columns interchanged. For k < 0, see also Figure 7.

where Z := kψ2 + β + vβψ, c = ch0h−δE1 , and d = dh−δE

1 . Equation (3.10) yields a functionα = α(β, ψ), valid only for Z > 0. Since δE > δE , (3.10) implies that

(3.11) α(β, ψ) = −cZδE + O(Z δE

)and hence that the sign of α for which P -homoclinics exist is that of −c. Homoclinic tangenciesbetween W u(P ) and W s(P ), for which the manifolds are in the configuration shown in themiddle column of Figure 6, must satisfy ∂α/∂ψ = 0, i.e.,

(3.12) −cδEZδE−1Z ′ = 0,

where Z ′ ≡ dZ/dψ = 2kψ + vβ. Thus for Z > 0 homoclinic tangencies occur at ψ = ψ ≡−vβ/2k. At this point Z = β to leading order, and it follows from (3.11) that such tangenciesoccur, to leading order, along the curve

(3.13) α = −cβδE , β > 0,

in the (α, β)-plane emanating from the EP1t-point. On this curve, the sheet of P -homoclinicsparametrized by α and β has a smooth fold.

To determine the location of the nearby P -homoclinics, we fix a value of α and defineβ = β(α) to be the value of β at which the homoclinic tangency occurs. Using a Taylorexpansion, (3.10) can now be rewritten, for |β − β| � 1, |ψ − ψ| � 1, and β � 1, in the form

(3.14) η = δE cβδE−1[k(ψ − ψ)2 + β − β],


(a)

z

Σ0

y

W u(P )

(b)

Σ1

x

y

W u(E)

W u(P )

(c)

ψ

η

Σ2

W u(E)W u(P )

(d)

ψ

η

Σ2

W u(E)

W u(P )

Figure 7. Case I, k < 0: Sketch of the images on various cross-sections of W u(P ) (bold curve) and of theline {y = y + rβ} ⊂ Σ0 to which W u(P ) ∩ Σ0 degenerates for g = 0 (thin line). The large dots in panel (a)mark points where Z = 0, which are heteroclinic connections from P to E. In panels (b)–(d), the large dotsshow the intersection of W u(E) with the cross-section, which are approached as Z → 0, and in (c) and (d)η = α. Panels (c) and (d) show different possible locations and orientations of W u(P ) in Σ2. Here boxesdenote extremal points in the η-direction which produce folds of P -homoclinics when they cross the ψ-axis.

again to leading order. It follows that W u(P )∩Σ2 is oriented in the manner shown in Figure 6.For example, for c > 0, k > 0, and β < β, W u(P ) has a local minimum that falls below η = 0and a pair of homoclinics is present. These are destroyed at the tangency as β increasesthrough β. For c < 0, k > 0, and β < β, W u(P ) has a local maximum which lies aboveη = 0 and a pair of homoclinics is again present; these are destroyed at the tangency as βincreases. The corresponding results for other combinations of the signs of c and k can beeasily computed.

It is clear from (3.10) that, if c �= 0, P -homoclinics can exist for α < 0 or α > 0 (if c > 0or c < 0, respectively) but not both. Assuming c > 0 and k > 0 and examining (3.11) in thelimit α → 0 from below and β > 0 fixed, we see that Z → 0 from above; i.e., the P -homoclinicorbits approach the heteroclinic connection from P to E. At α = 0, P -homoclinics cease toexist when they collide with W s(E) and so cannot escape from E to return to a neighborhoodof P . In parameter space, this occurs on the half of the β-axis where EP1 cycles occur (β > 0);along this half-line, two sheets of P -homoclinics meet in a cusp. Similar arguments apply toother combinations of the signs of c and k.

The case k < 0 can be analyzed further, because then W u(P ) ∩ Σ2 is a closed curve thatshrinks to a point as β → 0. The shape of W u(P ) where it intersects various cross-sections isshown schematically in Figure 7. Panel (b) depicts the nondegenerate linear image of W u(P )within Σ1, which is the union of two graphs y(x), each of which is the union of one convex andone concave piece. P -homoclinics occur when the closed curve in panel (c) or (d) of this figure


β β

ββ

α α

αα

k < 0

k > 0 k > 0

k < 0

c > 0 c < 0

c < 0c > 0

Figure 8. Case I: Partial bifurcation set for P -homoclinic orbits for δE < 1. Each panel shows the locus ofhomoclinic tangencies (α = −cβδE , to leading order), where the sheet of P -homoclinic solutions folds smoothly(inset in upper left panel), and the part of the β-axis where this sheet ends at EP1-cycles in a line of cusps(inset in upper left panel). The shading indicates to which side of the tangency or cusps the sheet extends, i.e.,where the transverse homoclinic orbits exist.

intersects the ψ-axis. The large dots in panels (c) and (d) mark the intersection of W u(E)with Σ2; these points are approached along W u(P ) ∩ Σ2 as Z → 0. Boxes mark extremalpoints in the η-direction, which produce folds of P -homoclinics as they cross the ψ-axis. Dueto the convex-concave shape of W u(P ) ∩ Σ2, the number of such points cannot exceed threeand depends on the orientation of W u(P )∩Σ2 (which depends on Π1) and its detailed shape(which depends on β). Depending on these details, which we do not analyze here, the aboveleading-order result may be augmented by two additional curves of folds in the (α, β)-plane,with the curves possibly ending at a cusp point.

In summary, there are four possible (leading-order) bifurcation sets for P -homoclinic or-bits, as shown in Figure 8. With regard to the results on nonwandering sets in [25], the regionof P -homoclinic orbits lies in the region for which the nonwandering set is nontrivial andcontains the region for which there exists a “basic set.” The latter is an invariant hyperbolictransitive set coinciding with the closure of its periodic orbits (compare Figure 8 for c < 0,k > 0, δE < 1 with Figure 2 in [25]).

4. Case II: Real eigenvalues at E, negative Floquet multipliers at P . The underlyingassumptions for this case are the same as in Case I, except that the Floquet multipliers are− exp(−lT ) and − exp(mT ).

In R3, either both Floquet multipliers of P are negative or neither of them is negative.

In our construction of the map Π3, in the case that both Floquet multipliers are positive, weassumed that, near tangency, orbits crossing Σ3 with ξ > 0 last crossed Σ2 with η > 0. Thisis not necessarily the case if both Floquet multipliers of P are negative, since some of theorbits crossing Σ3 will have crossed Σ2 with negative η, traversing P an odd number of timesbefore crossing Σ3. We need to modify our construction of the local map Π2 to allow for this


L

Lη

ξ

η

ξΣ+

3

Σ−3

Σ+2

η

Σ−2

ξ

ψ

(a) (b)

Figure 9. Illustrating the dynamics in a neighborhood of P in the case of negative Floquet multipliers.(a) The image of a section defined by {ψ = const} under the flow once around P . (b) The relation of the signsof η in Σ+

2 ∪ Σ−2 and ξ in Σ+

3 ∪ Σ−3 . Here η < 0 in Σ−

2 and ξ < 0 in Σ−3 : Π2 maps Σ−

2 to Σ−3 for an odd

number of windings about P and maps Σ+2 to Σ+

3 for an even number.

possibility. The other maps are not affected by the sign of the Floquet multipliers.A convenient global neighborhood of P in the case of negative Floquet multipliers is a

twisted tube whose normal cross-sections are rectangles, each with a center point on P atsome phase ψ. Such a tube would have only two surfaces, with the “top” and “bottom” ofeach rectangle lying on the same surface and the “left” and “right” sides both lying on theother surface. However, since we are interested only in orbits close to the heteroclinic orbitwith asymptotic phase zero at P , the neighborhood of P and corresponding cross-sectionsneed to be defined only locally, i.e., near ψ = 0.

To construct a modified local map near P , we have to keep track of changes in the sign ofthe η-component, as mentioned above. Sign changes in η within the ingoing cross-section givecorresponding sign changes in ξ within the outgoing cross-section. See Figure 9. Therefore,we define the map Π2 : Σ2 → Σ3, where Σ2 is the preimage of Σ3 in Σ2 under the backwardflow, by

Π2(ψ, η, h2) =

(ψ − Ω

mln(|η|h3

)− 2nπ , h3 , sgn(η)h2

(|η|h3

)δP)

.

The map is not defined if η = 0. The quantity n is a positive integer and counts the numberof complete circuits of P made by an orbit in passing from Σ2 to Σ3; it is even for η > 0 andodd for η < 0.

Calculations similar to those in section 3.2 yield the loci of E-homoclinic bifurcations.These occur, at leading order, when

Ψ − Ωm

ln(|α|) = 2πn − vβ

2k±√

S

ksgn(α)|α|δP − β

k.

Turning points in the E-homoclinic locus occur when the discriminant vanishes, i.e.,

β = sgn(α)S|α|δP .

Successive turning points occurring at (αn, βn) are naturally enumerated by integers n ≥ n0

for some large positive n0 in such a way that odd n is equivalent to negative α. In terms of


S < 0k > 0

S < 0k < 0

S > 0k < 0k > 0

S > 0

α α

αα

ββ

β

(αn+2, βn+2)

β (αn, βn)

(αn+1, βn+1)

Figure 10. Case II: E-homoclinic orbits occur locally on the solid curves. All panels show the case δP > 1;the case δP < 1 differs only in the shape of the dashed curve β = sgn(α)S|α|δP , which is the envelope of thefolds of the E-homoclinic loci.

the Floquet multipliers, we compute

αn = (−1)nKM−n, βn = (−1)nSKδP Ln,

so the signs of α and β both switch as n is incremented. The direction of the folds is determinedby the sign of the coefficient of β in the discriminant, i.e., by the sign of −k, as in section 3.2.The corresponding bifurcation sets are sketched in Figure 10.

The locus of the homoclinic bifurcations of P will be unaffected by this modification of themap Π2 (since Π2 was not used in our calculations) and will remain as computed in section 3.3,although the dynamics associated with the homoclinic bifurcations (e.g., the bifurcations ofnearby periodic orbits) may change.

5. Case III: Complex eigenvalues at E, positive Floquet multipliers at P . The assump-tions for this case are those of Case I except that the Jacobian matrix of the flow evaluatedat E has eigenvalues μ, −λ ± iω with distinct μ > 0, λ > 0, and ω a positive constant.

The local map Π0 needs to be modified to cover the case where the Jacobian matrix ofthe flow evaluated at E has complex eigenvalues. This modification will affect the locus ofP -homoclinic tangencies but not of E-homoclinic bifurcations (which will remain as calculatedin section 3.2). The behavior of periodic orbits associated with the homoclinic bifurcations ofboth E and P may be affected. The calculation that follows is an adaptation of that in [16]for the Shil’nikov–Hopf bifurcation.

To obtain the modified version of Π0, we use local polar coordinates (r, θ, z) near E chosenso that the flow near E can be written as

(5.1) r = −λr, θ = ω, z = μz.


z

Σ0

θ

β

ψ

β

α

η

Σ2

(a) (b)

Figure 11. Case III with k < 0, β > 0: Sketches of (a) W u(P ) ∩ Σ0, and (b) W u(P ) ∩ Σ2. The casein (b) is near a side-tangency; spiraling continues into the point (α, fα), which is the image of W u(E). Theindicated direction of tip motion/rotation is for increasing β.

The nonresonance assumptions on λ and μ guarantee that this is possible. Since we are usingpolar coordinates, it is convenient to define a new cross-section

Σ0 = {(r, θ, z) | r = h0}

for h0 a small positive constant. The cross-section Σ1 defined earlier does not require modi-fication. For sufficiently small h0 and h1 the flow then induces a map from Σ0 to Σ1. Using(5.1), we find to lowest order

Π0(h0, θ, z) =(

hzδE cos(

θ − ω

μln(

z

h1

)),

hzδE sin(

θ − ω

μln(

z

h1

)), h1

), z > 0,

where h = h0h−δE1 , and

δE =λ

μ

plays a role equivalent to that in the real eigenvalue case.The cross-section Σ0 is also used in the derivation of the global map Π3. Arguing as in

our derivation of Π3, we compute the modified global map Π3 : Σ3 → Σ0, which approximatesthe dynamics near a heteroclinic tangency between W u(P ) and W s(E). To lowest order in ψ,ξ, and β, we find

Π3 : (ψ, h3, ξ) → (h0, θ, z)(5.2)

= (h0 , gψ + qξ + rβ , kψ2 + sξ + β + vβψ),(5.3)

where g, q, r, k, s, v are constants with k �= 0. In calculating this map, we have chosen theorigin of the θ coordinate near E so that the heteroclinic tangency intersects Σ0 with θ = 0.

To locate P -homoclinics we first appeal to the geometry of W u(P )∩Σ2 shown in Figures11, 13, and 14 and schematically in Figure 12. The image of W u(P ) in Σ0 is a parabola by as-sumption, whose extremum occurs when the ψ-coordinate (in Σ3; ψ parametrizes W u(P )∩Σ3)


tip−tangencies

side−tangencies

β

αη

ψ

η

ψψ

ηη

ψ

Figure 12. Case III with β > 0 and k < 0 (for k > 0 reflected about the α-axis): A sketch of the typesof homoclinic tangencies in the (α, β)-plane in a region of parameter space in which curves of tip- and side-tangencies approach one another. Bold curves form the local parameter set of tangencies: tip-tangencies (green)and side tangencies (red). Insets show the relevant part of W u(P ) ∩ Σ2, with the dashed line being the centerof the parabola onto which W u(P ) ∩ Σ2 collapses for g = 0. The grey shading indicates the region in which itis not useful to distinguish between the two types of tangency.

(a)

z

Σ0

θ

β

(b)

Σ2

ηβ

ψ

Figure 13. Case III with k > 0, β > 0: Sketches of (a) W u(P ) ∩ Σ0, and (b) W u(P ) ∩ Σ2 near P at atip-tangency. The indicated direction of tip motion/rotation is for increasing β.

(a)

z

θ

Σ0

β

(b)

ψ α

Σ2

η

Figure 14. Case III with k > 0, β < 0: Sketches of (a) W u(P ) ∩ Σ0, and (b) W u(P ) ∩ Σ2 near P .Spiraling continues into the point (α, fα) which is the image of W u(E) and the bullets plotted in (a).

takes the value ψ = −vβ/2k. The z-coordinate of the extremum is O(β). The image in Σ1

of this parabola bounds a spiral-shaped region with a round tip. An affine map takes thespiral from Σ1 to Σ2, preserving its topological properties, so that the parameter β winds orunwinds the spiral region and controls its thickness, while α moves the core of the spiral in


the (η, ψ)-plane (in Σ2) in the direction (1, f). The parameter k determines the direction offolding at the spiral tip. Thus, for k > 0, β > 0, the tip is the inner end of the spiral, and thespiral has finitely many windings (see Figure 13), while for k < 0, β > 0 the tip is the outerend of the spiral and the spiral has infinitely many windings and is centered on the point(α, fα) (see Figure 11). For k > 0, β < 0 the spiral again has infinitely many windings, butthe tip is now at (α, fα) at the core of the spiral (see Figure 14). In the case k < 0, β < 0there are (locally) no intersections between W u(P ) and Σ2.

Intersections of W u(P ) ∩ Σ2 with η = 0 correspond to P -homoclinic orbits, which arecreated or destroyed as either the spiral tip or one of its sides crosses the line η = 0. We dis-tinguish between tip-tangency, which is primarily controlled by varying β, and side-tangency,which is primarily controlled by varying α (see Figures 12 and 13). The geometry of thesetangencies is shown in Figure 12. The distinction between tip- and side-tangencies becomesinvalid near places where the two types of tangency coincide, as will be discussed furtherbelow.

We investigate further the nature of P -homoclinic tangencies by analyzing the maps de-rived above. To locate the tangencies we compute Π1 ◦ Π0 ◦ Π3(ψ, h3, 0) and equate to zerothe η-component of this expression and also its derivative with respect to ψ. We find thatintersections between W u(P ) and W s(P ) occur, to lowest order, for

(5.4) α = α(ψ, β) = −cZδE cos A,

whereZ := kψ2 + β + vβψ, A := Φ + gψ + rβ − ω

μln Z,

and c = h0h−λ/μ1

√c2 + d2 > 0, Φ = tan−1(−d/c) + (ω/μ) ln h1. Expression (5.4) is valid only

if Z > 0. To find the P -homoclinic tangencies, we need to solve dα/dψ = 0 for α satisfying(5.4), which reduces to solving

(5.5) δEZ ′ cos A −(

gZ − ω

μZ ′)

sin A = 0

for ψ, where Z ′ ≡ dZ/dψ. We can understand the nature of the solutions to this equation byfirst examining the special case g = 0 and then incorporating small values of g. When g = 0,(5.5) reduces to

(5.6)1μ

Z ′(λ cos A + ω sin A) = 0.

The solutions with Z ′ = 0 correspond to tip-tangencies, while those with tan A = −λ/ω,Z ′ �= 0 correspond to side-tangencies. We examine these two possibilities in turn.

For the tip-tangencies, Z ′ = 0, we obtain ψ = −vβ/2k and hence Z = β + O(β2),A = Φ − ω

μ ln β + O(β). From (5.4) it now follows that tip-tangencies occur when

(5.7) α = −cβδE cos(

Φ − ω

μln β

)+ O

(β1+δE

),


where the coefficient of the O(β1+δE ) term vanishes at v = r = 0. Thus the case 0 < g � 1corresponds to a small perturbation of the snaking curve arising in Shil’nikov dynamics [13].At this order, the sign of k > 0 does not affect the position of tip-tangencies, although it doesdetermine on which side of these tangencies (in the (α, β)-plane) homoclinic solutions exist.

In contrast, for the side-tangencies we obtain

(5.8) Φ + rβ − ω

μln Z = − arctan

(λ

ω

)+ nπ,

where by arctan we mean the principal value in (−π/2, π/2], and n > 0 measures the numberof multiples of π required to reach equality in this expression; i.e., n measures the number ofoscillations the tangent P -homoclinic orbit makes close to E. Hence

(5.9) Z = Zn ≡ Anerβμ/ω ,

where An := e(Φ−nπ)μ/ω decays exponentially as n → ∞ and Φ = Φ + arctan(λ/ω). ThusZn → 0 exponentially with n. Substituting (5.9) into (5.4) now yields

(5.10) α = αn = −cAδEn erβλ/ω cos (arctan(λ/ω) − nπ) ,

yielding a sequence of α values of alternating sign converging exponentially to zero. Since βis small, (5.9) can now be solved to obtain β = βn(ψ), where

(5.11) βn = −kψ2 − vAnψ + An + O(rAnψ2 + vψ3 + rA2n).

Thus the curves of side-tangencies in the (α, β)-plane are parametrized by the original ψ-coordinate (in Σ3) and are, at leading order, parabolas with curvature given by −k. Turningpoints of these curves relative to the β-direction occur (to leading order) at ψ∗ = −Anv/2kand β∗

n = An. Substituting into (5.4), we find that at leading order

α∗n = −c(β∗

n)δE cos(

Φ − ω

μln β∗

n

),

so these points lie on the leading-order curve of tip-tangencies given by (5.7).The higher order terms in (5.7), (5.10), and (5.11) vanish when v = r = 0. In this case

the tip-tangency and side-tangency curves are globally connected, and the tip-tangency curvereduces to the snaking curve

α = −cβδE cos(

Φ − ω

μln β

),

while the parabolas of side-tangencies collapse onto the vertical half-lines

{α = −ce(Φ−nπ)λ/ω cos(arctan(λ/ω) − nπ), sgn(β − e(Φ−nπ)μ/ω) = sgn(−k)};

i.e., the turning points of the tip-tangency curve in the α-direction coincide with the turningpoints of the doubly covered vertical side-tangencies in the β-direction, as shown in Figure


(a) α

β

tip-tangency

side-tangency

(b)

~~

(i) (ii)

rv

g

Figure 15. Case III: Sketch of the curves of P -homoclinic tangencies in the case k > 0. (a) The leading-order geometry, i.e., the special case g = r = v = 0. (b) A schematic for the case g, r, v �= 0 showing thepossible connections of the different tangency curves. The inset at the top shows the four curves emanatingfrom the crossing point; note that the vertical curve is double. Since two of the four curves coincide in thedegenerate case, there are only two possible connectivities: (i) tip- to side-tangency, (ii) tip- to tip-tangencyand side- to side-tangency. The precise configuration within each of the shaded circles and the values of r, g, vcorresponding to each case are beyond the scope of our analytical considerations.

(a) alpha

beta

-0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

α

tip-tangency

side-tangency4 sol’s

2 sol’s

0 sol’s

β

(b)-0.020

-0.0100.000

0.0100.020

0.0300.040

0.0500.060

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

α

β

(c)-0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

α

tip-tangency

4 sol’s

2 sol’s

0 sol’s

β side-tangency

(d)-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

α

β

Figure 16. Numerical solutions to the exact P -homoclinic tangency conditions (5.4) and dα/dψ = 0 forc = −1, v = 1, δE = 1/2, ω/μ = 1, Φ = 0, g = 1, r = 1. (a) The case k = 1. Labeling of tangencies is interms of the limiting leading-order approximations; numbers refer to relative change in number of P -homoclinicsolutions. (b) Magnification of (a). Note how the disconnection between the side and tip tangencies becomesless pronounced as α, β → 0. (c) Similar depiction of the case k = −1. Panel (d) is the same bifurcation setas in (c) but for a larger range of α and β.

15(a). Upon perturbing g, r, v away from zero, we expect the doubly covered vertical side-tangency curves and their connections at the turning points to separate (Figure 15(b)). Indeed,direct numerical computation of the condition for double roots of (5.4), shown in Figure 16,corroborates the geometric observation that can be inferred from Figure 12: the two pairs oftangency curves meeting at turning points cross-connect in one smooth curve and one cusp.


Specifically, we find that for k > 0, the outward side-tangency and upper tip-tangency curvesform a smooth curve, and the inner side-tangency and lower tip-tangency curves form a cusp.The case k < 0 creates the same connectivity but with the direction of β reversed (see Figure16(c)).

We note one curiosity in panel (d) of Figure 16 for the case k < 0: the curves of side-tangencies reconnect for large β producing a Calla lily–type structure. This reconnection doesnot happen for k > 0. Presumably this is a consequence of the topological distinction betweenthe spirals W u(P ) ∩ Σ2 which have infinitely many windings for small positive β when k < 0but only finitely many for positive β when k > 0, with this number tending to zero as βincreases. However, this behavior is outside the scope of our local analysis and most likelyirrelevant to the unfolding of the bifurcation in question for which we implicitly assume thatα and β are small.

In summary, the transition from real to complex eigenvalues affects neither the locationsof the heteroclinic connections, cycles, and tangencies (these are as shown in Figure 4 for thecase k > 0) nor the location of the E-homoclinics. However, the location of the P -homoclinicsis substantially changed. For real eigenvalues the parameter regime for which there are P -homoclinic orbits is bounded by two curves—C1, the half of the β-axis where EP1-cyclesexist, and C2, a curve emanating from the origin and continuing monotonically in α. Asthe eigenvalues become complex, the analogue to C2 folds infinitely often about C1 (so thatC2 becomes a snaking curve) with additional, almost vertical curves of tangencies appearing.Pairs of P -homoclinics are created on each curve of P -homoclinic tangencies, and so there areregions of parameter space in which there are no P -homoclinics, other regions where there aretwo P -homoclinics, other regions where there are four P -homoclinics, and so on, as illustratedin Figure 16.

6. Dynamics of the full return map. A complete unfolding of the dynamics in the neigh-borhood of an EP1t-point is likely to be highly involved, especially in the case of complexeigenvalues at E satisfying the Shil’nikov condition δE < 1. Even in the absence of a P -homoclinic tangency, it is known that near such an equilibrium, shift dynamics occurs withrespect to arbitrarily many symbols, and the dynamics is nowhere structurally stable, forexample, because of arbitrarily close parameter values where multipulse E-homoclinics oc-cur [14]. For the P -homoclinic tangencies, so-called wild hyperbolic sets (or Newhouse sinks)necessarily occur at nearby parameter values, with (if δP < 1) parameter values with infinitelymany stable periodic orbits. See, for example, [27, 12, 10]. The combination of these two kindsof dynamics will be yet more complex.

We focus, therefore, on answering the question of what happens to the simplest foldsof primary periodic orbits in a neighborhood of the EP1t-point. By primary, we mean thoseperiodic orbits that form a sequence which converges on the single-pulse E-homoclinic orbits orP -homoclinic tangency. In both cases, where this is an accumulation of folds on the homoclinicorbit in question, there is also an accumulation of period-doubling bifurcations. However, fortopological reasons, these period-doublings typically occur very close to the fold bifurcations.See, for example, [13, 11]. So, for simplicity, we do not compute the loci of period-doublingbifurcations in the parameter plane, as, broadly speaking, we can infer this information fromknowledge of the location of the folds. We consider positive Floquet multipliers of P only.


6.1. Case I: Real eigenvalues of E, positive Floquet multipliers of P . In this case, awayfrom the EP1t-point, there are no infinite sequences of folds of periodic orbits accumulating onthe E-homoclinic orbits, so the question becomes: What happens as we approach the EP1t-point to the sequence of folds of periodic orbits known to accumulate on the P -homoclinictangency?

We start by constructing the full Poincare map in this case. Specifically we construct themap R : Σ2 → Σ2, R ≡ Π1 ◦Π0 ◦Π3 ◦Π2. Upon combining expressions (3.2), (3.4), (3.5), and(3.6), we obtain

(6.1)

⎛⎝ ψηh2

⎞⎠ →

⎛⎜⎝ aZδEn + b(y + gΛn + qηδP + rβ)Z δE

n + fα

cZδEn + d(y + gΛn + qηδP + rβ)Z δE

n + αh2

⎞⎟⎠ ,

where

Zn(η, ψ, β) = kΛ2n − SηδP + β + vβΛn, Λn(η, ψ) = Ψ + ψ − Ω

mln η − 2nπ.

Here the integer n counts the number of complete circuits an orbit makes close to P , S =−sh2h

−δP3 as before, and the various constants with tildes are defined in terms of the original

constants as

a = ah0h−δE1 , b = bh−δE

1 , c = ch0h−δE1 , d = dh−δE

1 , q = qh2h−δP3 ,

with Ψ = (Ω/m) ln h3. Note, from the construction, that the map is valid only for

η > 0 and Zn > 0.

Periodic orbits of the flow correspond to fixed points of map (6.1).The condition for a fixed point can be considerably simplified by considering only the

leading-order terms of (6.1), remembering that δE > δE and that the terms being raised tothese powers must be small for a fixed point at small α and β to be present. Hence we havethe leading-order fixed point conditions:

ψ = aZδEn + fα,

η = cZδEn + α.

The second equation can be used to eliminate Zn from the first equation, resulting in anexpression for ψ in terms of η and α. Substituting this into the second equation then gives

(6.2) η = cZδEn + α, η, Zn > 0,

where

Zn(η, α, β) = kΛ2n − SηδP + β + vβΛn, Λn(η, α) = Ψ + aη + fα − Ω

mln η − 2nπ,

and a = a/c, f = f − a/c.


(I) δE , δP < 1 (b) β = 0(a) β < 0

η

β

(II) δE , δP > 1 (d) β < 0(e) β = 0

α η

α > 0α = 0

α < 0

(f) β > 0

η

ηηη

(c) β > 0

Figure 17. Sketch of the right-hand side of (6.3) for different n (dashed curves) and the envelope as n → ∞(solid blue curve); the left-hand side of (6.3) for various values of α (solid red lines) is also shown. Here f = 0,k > 0, c > 0, and S < 0. (a)–(c) δE < 1, δP < 1; (d)–(f) δE > 1, δP > 1. Intersections between the red curveand dashed lines for positive ordinate and abscissa correspond to fixed points of the return map, and doubleroots correspond to fold bifurcations. Note that the folds occur at η ≈ ηn, where the dashed curve touches itsenvelope; the latter are indicated by solid circles.

Fold bifurcations are given by double roots of the fixed point equation for the map (6.1),and hence to leading order by double roots of (6.2). It is instructive to rewrite (6.2) in theform

(6.3)(

η − α

c

)1/δE

= kΛ2n − SηδP + β + vβΛn

and plot the left- and right-hand sides as η varies. In doing so, we note that the right-hand sidehas local extrema for Λn ≈ 0 (one for each choice of n), and these occur when the right-handside takes the approximate value β − SηδP .

For now we make the simplification f = 0 and consider the case δE < 1, δP < 1 with thechoice k > 0, c > 0, S < 0, as depicted in Figure 17(a)–(c). For β > 0 fixed there are infinitelymany values of α for which the graph of the left-hand side of (6.3) (the red curve) is tangentto the graph of the right-hand side (the dashed black curve). These values are fold points,and all occur for negative values of α and converge on the value β = (−α/c)1/δE , which is thecondition α = −cβδE derived earlier for P -homoclinic tangencies. A tangency between thegraphs involving one particular loop of the graph of the right-hand side (i.e., one particularvalue of n) can be maintained by simultaneously decreasing β while increasing α, and we thusfind curves of folds in the second quadrant of the (α, β)-plane accumulating on the curve ofP -homoclinic tangencies.


E−hom

per−fold

2 01

β

α

Figure 18. Schematic diagram showing a partial unfolding of the inclination flip bifurcation in the (α, β)-plane. The curve labeled per-fold is the locus of a fold of periodic orbits. The arrows indicate the direction ofbifurcation of periodic orbits from the E-homoclinic; this direction changes at the point where the fold touches theE-homoclinic curve, i.e., at the inclination flip point. The numbers indicate how many periodic orbits exist ineach region of parameter space. Note that the unfolding is partial because there may exist other bifurcations thatdo not affect our analysis, i.e., multipulse homoclinic bifurcations, period-doubling bifurcations, and horseshoedynamics.

For fixed β < 0, there are only finitely many tangencies between the two graphs for positiveZn; the tangency corresponding to a particular choice of n will persist as β is decreased throughzero only until the corresponding loop of the graph of the right-hand side of (6.3) first touchesthe η-axis. This occurs when Λn ≈ 0, i.e.,

ηn ≈ emΩ

(Ψ−2nπ),

and so the last tangency for this n occurs for α ≈ ηn (= αn as defined in section 3.2)and β ≈ SηδP

n (= βn as defined in section 3.2), i.e., close to the tip of the correspondingparabola of E-homoclinic orbits in the fourth quadrant of the (α, β)-plane (see Figure 5).Numerical solutions of (6.2), shown in Figure 19(a), confirm these conclusions: each branchof folds terminates when it reaches the tip of an E-homoclinic parabola. As such a point isapproached, Λn → 0, and so Zn → 0, implying that the period of the periodic orbit undergoingthe fold diverges to infinity as it approaches the E-homoclinic bifurcation. The point wherethe fold and the E-homoclinic touch will thus be a codimension-two point (Figure 18) at whichthere is a switch in the direction (in parameter space) in which the periodic orbit emergesfrom the homoclinic orbit. To see that this must occur, note that there are three types ofbifurcations of homoclinic orbits that lead generically to this type of side-switching. One ofthese occurs as a result of eigenvalue resonance [5] (λ = μ in our notation), while another isthe so-called orbit flip [31], in which the E-homoclinic orbit switches between components ofthe leading stable eigenvector. The latter cannot happen here owing to our assumptions onthe geometry near E. Thus the side-switching must be due to an inclination flip. At sucha point the stable manifold of E followed (backward in time) along the E-homoclinic orbitchanges between being orientable and nonorientable [18, 19].

There are, in fact, three types of unfolding for the inclination flip bifurcation, dependingon the eigenvalues. Following the notation of [17] these are the cases A (no other bifurcations),


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5

(a)

(c) (d)

(b)

n = 1n = 5

β

ββ

β

α

α

α

α

Figure 19. Curves of fold bifurcations of periodic orbits (solid red lines) in the case of real eigenvalues forE, computed using the one-dimensional map (6.2) for 1/Ω = 0.1, f = a = 1/2, c = 1, ψ = 0, k = 1, S = −1,v = 0 and (a) δE = δP = 1/2, (b) δE = 1/2, δP = 3, (c) δE = 3, δP = 1/2, and (d) δE = δP = 3. Short (blue)and long (green) dashed lines represent, respectively, the position of E-homoclinic orbits and P -homoclinictangencies, given by formulae (3.8) and (3.13), respectively.

B (homoclinic doubling), and C (complex dynamics including n-homoclinic and n-periodicorbits for arbitrary values of n and horseshoe dynamics). Case A occurs only for δE > 1,which is not the case under investigation here. The other two cases occur for δE < 1; whichof these occurs depends on the value of the nonleading stable eigenvalue. However, both casesfeature the fold bifurcation and side-switching, and the extra bifurcations do not concern us.

We next consider the case δE > 1, δP > 1 with the same choice of signs of k, S, and c,depicted in Figures 17(d)–(f) and 19(d). Again, for β > 0, there are infinitely many values ofα for which there are folds, and these converge on α = −cβδE . However, in contrast to theprevious case, there are now infinitely many tangencies between the two graphs for arbitrarilylarge |β|, with β < 0. This is a consequence of the infinite slope of the left-hand side of (6.3)as a function of η as η − α → 0. This curve can now become tangent to the dashed lineat points other than those on the blue envelope where η = ηn. Such a point of tangency,which must occur for α > 0, is depicted in Figure 17(d). Note that as β → −∞ these pointsoccur closer and closer to Zn = 0, and so α becomes closer and closer to the η value at whichthe right-hand limb of the corresponding dashed curve intersects the η-axis in Figure 17.These are the parameter values for the right-hand part of the parabolic E-homoclinic curvesin the (α, β)-plane. Thus each (red) curve of folds closely hugs the right-hand branch of thecorresponding (blue) curve of E-homoclinic orbits in Figure 19(d).


The cases δE < 1, δP > 1 and δE > 1, δP < 1 can be analyzed similarly. Numericalsolutions of (6.3) are shown in Figure 19(b) and (c), respectively. Note that the crucialdistinction is in the sign of δE − 1. If this is positive, we find inclination flip points near thetips of the E-homoclinic parabolas, while if it is negative, we find persistence of folds to largenegative β.

The simplifying assumption f = 0 is not, in fact, necessary. In particular, a Taylorexpansion with respect to η can be used about each point at which (6.3) has a double rootη∗ with respect to η to show that a rescaling of α by the multiplicative factor (1 + fη∗m/Ω)leads to an equation that is identical to leading order but with f = 0.

We have focused here on the case k > 0, c > 0, S < 0. Cases with different signs of k, c,and S can be analyzed similarly.

6.2. Case III: Complex eigenvalues at E, positive Floquet multipliers at P . In contrastto the previous sections, we do not give a rigorous analysis of this case. Instead, we are guidedby numerical results and topological arguments.

The full Poincare map in this case is R : Σ2 → Σ2, R ≡ Π1 ◦ Π0 ◦ Π3 ◦ Π2. Composingthe maps written before and using the compound angle formula to simplify the expressionsobtained, we find

(6.4)

⎛⎝ ψηh2

⎞⎠ →

⎛⎜⎜⎝AZδE

n cos(θ1 + gΛn + qηδP + rβ − ω

μ ln Zn

)+ fα

CZδEn cos

(θ2 + gΛn + qηδP + rβ − ω

μ ln Zn

)+ α

h2

⎞⎟⎟⎠ ,

where

Zn(η, ψ, β) = kΛ2n − SηδP + β + vβΛn, Λn(η, ψ) = Ψ + ψ − Ω

mln η − 2nπ,

A = h0h−δE1

√a2 + b2, C = h0h

−δE1

√c2 + d2, S = −sh2h

−δP3 , q = qh2h

−δP3 ,

θ1 = tan−1

(− b

a

)+

ω

μln h1, θ2 = tan−1

(−d

c

)+

ω

μln h1,

and all other quantities and constants are as defined after (6.1). The map is valid only for

η > 0 and Zn > 0.

Equations (6.4) can be simplified slightly by noting that we are interested in solutionswith Zn small, in which case two terms in the argument of each of the cosine terms in themap can be ignored. The simplified map is

(6.5)

⎛⎝ ψηh2

⎞⎠ →

⎛⎜⎜⎝AZδE

n cos(θ1 + gΛn − ω

μ ln Zn

)+ fα

CZδEn cos

(θ2 + gΛn − ω

μ ln Zn

)+ α

h2

⎞⎟⎟⎠ ,


-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

-0.1 0 0.1 0.2 0.3 0.4

-0.2

-0.1

0

0.1

0.2

-0.1 0 0.1 0.2 0.3 0.4

2D

1D

4 3 2 1

2 1

5

P

E

P

35

4

E

54

3

1

2

P

E(b)

(d)

(a)

(c)

Figure 20. Curves showing fold solutions to the approximate two-dimensional map (6.5) for fixed parameters(6.6) (a) g = 0, ω/μ = 2, δE = 3; (b) g = 1, ω/μ = 1, δE = 0.5. Each curve of folds is labeled by the numbern of winds near P made by the corresponding periodic orbit. Also plotted in panels (a)–(c) are the loci ofE-homoclinic orbits (blue dot-dashed curves) and P -homoclinic tangencies (green dotted curves) given by theformulae derived earlier. Panel (c) is a blow-up of (b) and (d) shows the same fold data (in red) plottedtogether with the corresponding fold curves of the one-dimensional approximation (6.7) to the map.

whereZn(η, ψ, β) = kΛ2

n − SηδP + β.

We seek double roots of the fixed point equations for this map which correspond to folds ofperiodic orbits of the underlying flow. Figure 20 shows a detailed computation of fold pointsof this two-dimensional map in the (α, β)-plane for the parameter values

A = 0.5, C = 1, f = 1, k = 1, S = −1, δP = 3,θ1 = 1, θ2 = 0, Ψ = 0, Ω/m = 10

(6.6)

and different choices of the values of the remaining parameters.The main distinction between the first two panels in Figure 20 is the value of δE : in

panel (a) δE > 1, whereas in panel (b) (which is magnified in panel (c)) δE < 1. In panel (a),according to the theory of Shil’nikov homoclinic bifurcations [11, 13], each E-homoclinic orbitshould be surrounded by at most finitely many fold bifurcations. This is borne out in panel (a),where we see an additional pair of folds joined by a cusp near each curve of E-homoclinics. Incontrast, in panel (b) we find strong numerical evidence for infinitely many fold curves near


each E-homoclinic locus. Only a few curves (representative consecutive folds along a branchof periodic orbits approaching each E-homoclinic orbit) are shown. Note the presence ofadditional cusps as well. We have performed other computations of fold curves of these mapsfor different values of the coefficients. The same broad features are obtained. However, theprecise location of, for example, the cusp points depends sensitively on the values of variousparameters, such as c, k, S, and g.

In order to explain the characteristic shape of these fold curves and the presence of cuspson them, it is tempting to try to reduce (6.5) to a one-dimensional fixed point equation, justas we did in the case of real eigenvalues to produce (6.2). In that case, simple algebraicmanipulation led to the elimination of ψ. However, this approach does not work here unlessθ1 = θ2 since the two cosine terms in (6.4) have different arguments. On the other hand, ifone supposes that ψ must be small compared to Ω

m ln η in the expression for Λn, one is led tolooking at fixed points of the one-dimensional map

(6.7) η → CZδEn cos

(θ2 + gΛn − ω

μln Zn

)+ α,

where all terms are defined as before except for

Λn(η) ≡ ψ − Ωm

ln η − 2nπ.

A similar argument suggests that the term gΛn is then also small compared with ωμ ln Zn

and hence can also be ignored to leading order. With these assumptions, folds correspond todouble roots of the fixed point equation

(6.8) η = CZδEn cos

(θ2 −

ω

μln Zn

)+ α.

Note the similarity of (6.8) to the corresponding fixed point equation (6.2) in the real eigen-value case. In particular, the right-hand side of (6.8) is a rapidly oscillating function of ηwhose envelope is identical (up to definition of the constants) to the right-hand side of (6.2).This suggests that along curves of folds of the one-dimensional map in the real eigenvalue casewe should see “folds of folds” in this complex eigenvalue case or, in other words, cusps.

Unfortunately, all attempts to make the argument in the previous paragraph rigorous haveproved unsuccessful, and it appears that (6.5) is really a fully coupled two-dimensional map,much as in the case of a homoclinic orbit to a bifocus [9]. Nevertheless, numerically we findremarkably close agreement between the fold curves obtained by looking for fixed points ofthe two-dimensional map (6.5) and the folds of the approximate one-dimensional fixed pointequation (6.8), as demonstrated in Figure 20(d).

7. Homoclinic snaking and global connections of EP1t-cycles. In applications, we fre-quently find in two-parameter continuations that E-homoclinic orbits occur along a widesnaking curve like that shown in Figure 23(d); it appears that the E-homoclinic orbits areaccumulating on a pair of EP1t-points and on a curve in parameter space joining these points.In this section we give a geometric explanation for this phenomenon based on certain globalextensions of our Poincare map. Our approximate arguments are similar in spirit to the recent


(a) β

Norm

β∗

EP1t

EP1

(b)

β

β∗

α

Figure 21. (a) Sketch of a possible bifurcation diagram for connected EP1-cycles; large dots correspond toEP1t-cycles. (b) A possible extension of the situation in (a) to two parameters. In the shaded region there existtwo different heteroclinic connections from P to E, with tangencies occurring on the dashed line and the α-axis.Assuming a codimension-one connection from E to P exists for α = 0, there will be two distinct EP1-cyclesfor parameter values on the β-axis with β∗ < β < 0.

rigorous analysis by Beck et al. [2] for homoclinic snaking in the context of reversible systems,which has one less codimension.

We extend the model from the previous sections by making assumptions about propertiesof the heteroclinic connections from P to E as a parameter is varied over an O(1) intervaland about the ways in which the maps Πj behave over this interval. We start by notingthat in applications heteroclinic connections from P to E typically come in pairs. A pair iscreated at a tangency between W u(P ) and W s(E); the two connections persist for an intervalof some parameter and then annihilate pairwise at a second heteroclinic tangency. If a secondparameter can be tuned to select values at which codimension-one connections from E toP simultaneously exist, then we will get a curve of EP1-cycles in two-parameter space, withendpoints of the curve being EP1t-points. Figure 21 shows schematically how this might occurif we extend the parameters α and β, which locally unfold an EP1t-point, to global values.

Since we strive for a simple geometric explanation, we make the following assumptions.We assume that two EP1t-points exist, at (α, β) equal to (0, 0) and (0, β∗), where β∗ < 0,and that EP1-cycles occur at points on the β-axis between these two points so that we havethe situation shown in Figure 21. We further assume that P and E and their linearizationsremain unchanged in an O(1) parameter region containing this interval of EP1-cycles, as doesthe connection from E to P . Thus the maps Πj , j = 0, 1, 2, will remain valid and unchangedover this O(1) region. It is convenient to work with a cross-section, Σglob

3 , instead of Σ3,where Σglob

3 is defined near all phases of P rather than near just one specified phase. We canthen derive a new map, Πglob

3,β : Σglob3 → Σ0, that varies globally with β in a simple way, as

specified below. In the following, by W sβ(E), we mean the (β-dependent) curve of intersections

of W s(E) with Σglob3 if we run time backward, i.e., W s

β(E) = (Πglob3,β )−1 {(h0, y, 0)}. By W u

α (E)

we mean the curve on Σglob3 traced by the point of first intersection of W u(E) with Σglob

3 as αis varied (see Figure 23).

7.1. Positive Floquet multipliers at P . Using local coordinates near P as described insection 3, we define

Σglob3 = {(ψ, η, ξ) | η = h3, 0 ≤ ψ < 2π}.


(a)

bc

da

W u(P )

W s(E)

Σ0

z

y (b)

a b c d

ψ

W u(P )

W sβ(E)

0 2π

Σglob3

Figure 22. Geometry of W u(P ) and W s(E) on (a) Σ0, and (b) Σglob3 , when two EP1-cycles are present.

In Figure 22, we show schematically the simplest possible geometries of W u(P ) and W s(E)where they intersect Σ0 and Σglob

3 , for a β-value for which two EP1t-cycles exist. We assumethat, to leading order, the only effect of changing parameter β is to move W s

β(E) rigidly up (foran increase in β) or down (when β decreases) so that the two EP1t-cycles occur, respectively,when the points b and d on W u(P ) are simultaneously on W s(E). For simplicity, we assumethat W s

β(E) is sinusoidal in shape, but in fact any smooth periodic function with a singlemaximum and a single minimum will lead to the same result.

To find E-homoclinic orbits, we need to locate W u(E) on Σglob3 . Using maps Π1 and Π2,

we compute W u(E) ∩ Σglob3 to be at (ψ, h3, ξ) with

ψ =(

Ψ − Ωm

ln α

)mod 2π, ξ = hαδP ,

to lowest order, where h = h2h−δP3 . As α varies, this point traces W u

α (E), which is a sequenceof near-horizontal lines converging on W u(P ), as shown in Figure 23(a)–(c). E-homoclinicsoccur when W u

α (E) intersects W sβ(E). Since by assumption the position of the former does not

depend on β while the latter does not depend on α, the shape of the locus of E-homoclinicscan now be deduced as follows. We start with α and β values for which there is an E-homoclinic orbit (as indicated by the large dot in of Figure 23(a)) and decrease α monotonically(which moves the tracer point to the right) while simultaneously varying β to maintain theE-homoclinic orbit. Initially, the E-homoclinic orbit can be maintained by increasing β, whichshifts the sinusoidal curve up. However, once β is sufficiently large that the local minimumof W s

β(E) rises too high (this will occur for β ≈ 0, which is the case shown in Figure 23(b)),then β must be decreased to maintain the E-homoclinic orbit. A further small decrease inα can then be compensated for by decreasing β further, at least until the local maximum ofW s

β(E) falls too low (for β ≈ β∗, as in Figure 23(c)), in which case an increase in β will againbe required for maintenance of the E-homoclinic orbit. This process repeats, and we end upwith the wide snaking curve shown in Figure 23(d). A similar argument was used in [3], wherea snaking curve of E-homoclinic orbits, lying in an exponentially thin wedge, was found inthe neighborhood of a local codimension-two saddle-node/Hopf bifurcation point.

More formally, our global assumptions imply that W sβ(E) is a graph Fβ(ψ) : R → R with

Fβ(ψ+2π) = Fβ(ψ) and Fβ(ψ) = F0(ψ)+G(β) with monotone G. On the other hand, W uα (E)


(a)

ψ

0 2π

W u(E) W u(P )

Σglob3 W s

β(E)

(b)

ψW u(E)

0 2π

W sβ(E)

W u(P )

Σglob3

(c)

ψ

0 2π

Σglob3

W u(E)W sβ(E)

W u(P )

(d)

β α

(a)

β∗

(b)

(c)

Figure 23. Geometric explanation for the snaking E-homoclinic locus. W u(E) moves monotonically tolarger ψ as α decreases, tracing out the near-horizontal lines, while W s(E) is shifted monotonically up as βincreases. (a) Intersection of W u(E) and W s(E) at a value of β such as that indicated by label (a) in panel (d).(b) W u(E) intersects W s(E) at its local minimum for β ≈ 0. (c) W u(E) intersects W s(E) at its local maximumfor β ≈ β∗. (d) The corresponding E-homoclinic locus for the case S > 0. By assumption, k < 0 for the “lower”EP1t-cycle and k > 0 for the “upper” (cf. Figure 5). Labels indicate typical parameter values for the situationsshown in the corresponding panels.

is a parametrized curve in Σglob3 ,

α →((

Ψ − Ωm

ln α

)mod 2π , hαδP

).

E-homoclinic orbits occur whenever these curves meet, i.e., when

ψ =(

Ψ − Ωm

ln α

)mod2π and Fβ(ψ) = hαδP ,

which implies that

β = G−1

(hαδP + F0

(Ψ − Ω

mlnα

)).

Turning points in the E-homoclinic locus arise for α values for which

F ′0

(Ψ − Ω

mln α

)= 0,

i.e., near EP1t-cycles, as computed in section 3.The case discussed above occurs when there are two critical points for F , but there may

in fact be more (which would imply more than two connected EP1t-points). For instance, acase with six EP1t-points is shown schematically in Figure 24 and will be discussed furtherin section 8.


(a) β

Norm

β∗

EP1

(b)β∗

αβ

0 2π ψ

Σglob3

Figure 24. (a) Sketch of a possible bifurcation diagram in the case of six cyclically connected EP1t-points(represented by large dots). (b) Analogue of Figure 22(b) for case (a) traced counterclockwise, showing thecorrespondence between the folds in the multisnaking E-homoclinic locus and the critical points of F0. Forsimplicity, we do not show the geometry of the envelope of fold points in detail (cf. Figure 5).

7.2. Negative Floquet multipliers at P . In section 4 we found that the main effect ofnegative Floquet multipliers at P on the locus of E-homoclinic orbits was the simultaneoussign change of α and β at successive turning points, with the result that half the turningpoints accumulate on the EP1t-point from positive α and half accumulate from negative α(see Figure 10). The effect of negative Floquet multipliers on the snaking curve is analogous:there are now two snaking curves, accumulating on the interval of EP1-cycles from oppositesides of α = 0.

To see this, we first define an alternative version of Σglob3 appropriate to this case:

Σglob3 = {(ψ, η, ξ) | η = h3, 0 ≤ ψ < 4π}.

This is a double cover of all phases of P ; the need for this form of cross-section is as discussedin section 4. Proceeding as before, we again assume that W s

β(E) is a continuous graph withat least two critical points, as shown in Figure 25(a), while β moves it rigidly up or down. Asargued in section 4, an E-homoclinic occurring near a turning point close to one of the EP1t-points will cross Σglob

3 with ξ positive (resp., negative) if α is positive (resp., negative) at theturning point. Thus W u

α(E) now has two disjoint components, each consisting of a sequenceof near-horizontal lines accumulating on W s(P ) ∩ Σglob

3 , one with ξ > 0 and the other withξ < 0. Arguing as in the previous subsection for each component separately, we deduce thatthe locus of E-homoclinic orbits consists of two wide snaking curves, as described above andshown in Figure 25. The mechanism leading to the bifurcation of two snaking curves for thiscase is similar to the mechanism operating near E-homoclinic bifurcations from codimension-two equilibrium-to-periodic orbit heteroclinic cycles with winding number two, as describedin [28]: these cycles require a phase space of at least four dimensions and occur when theE-to-P connection is itself of codimension two.

8. Numerical examples. EP1t-points have been seen in various systems of ordinary differ-ential equations. A straightforward way to identify the EP1 phenomenon is via E-homoclinics.If homoclinic snaking or an accumulation of folds of E-homoclinic curves is seen in the bifur-cation set for a particular system, then we can diagnose the presence of the EP1 phenomenonby looking at phase portraits along the E-homoclinic curves using standard software such asAUTO and HomCont [6]; an E-homoclinic orbit will pick up extra loops near a periodic orbitand in fact will seem to converge to a periodic orbit as it gets close to a presumed EP1t-point.


(a)

Σglob3

4π0

W u(E) ψW sβ(E)

(b)

β α

β∗

Figure 25. (a) The analogue of Figure 23(a) for negative Floquet multipliers. The large dot shows theintersection of W u(E) with Σglob

3 for a typical positive value of α. The large square shows the same thing, butfor negative α. (b) The corresponding loci of E-homoclinic orbits. For simplicity, we do not show the geometryof the envelope of fold points in detail (cf. Figure 10).

It is also possible to find EP1t-points by continuing curves of P -homoclinic tangencies, hete-roclinic connections from E to P , and heteroclinic tangencies from P to E, but this involvesthe use of more complex software currently under development. See [7, 8, 21].

In section 8.1, we describe a system originally designed to model a saddle-node Hopfbifurcation with global reinjection [20]. The bifurcation set for this system contains a snakingE-homoclinic curve that converges in parameter space to a segment connecting two EP1t-points. We show that the rate at which E-homoclinic folds accumulate is consistent with thetheory in section 5 and that the bifurcation set has a structure of fold bifurcations and cuspsthat matches the predictions of section 6. In section 8.2, we discuss a nine-dimensional modelof intracellular calcium dynamics [34, 35]. The bifurcation set for this system also contains asnaking E-homoclinic curve, but there are now six EP1t-points involved, as in Figure 24.

Another example is provided by the FitzHugh–Nagumo equations, where we do not seea homoclinic snake but instead a sequence of folded curves of E-homoclinics that are notglobally connected in the parameter space. Numerics suggest there exist folds of periodicorbits accumulating on the E-homoclinic orbits and folds accumulating on a P -homoclinictangency, with the two families of folds being connected via cusps. Since the numericalevidence is not as clear here as for the previous two examples, and for reasons of brevity, wedo not discuss these equations further here but refer the reader to the discussion in [4].

The first of our example systems (saddle-node Hopf model) and the FitzHugh–Nagumoequations contain instances of Case III EP1t-points: complex eigenvalues at E and positiveFloquet multipliers at P . The model of calcium dynamics is too stiff for AUTO to be able tocalculate Floquet multipliers reliably, and so we are unable to determine which case we have.However, in this example, we have at least one EP1t-point at which the leading eigenvalue at


SNH S0 NSNS

H

Sl

ha/b0

ha1

hb1

hb1

DH

(a)ν1

ν2

-2

-1.9

-1.8

-1.7

-1.6

-1.5

-1.4

-0.4 -0.2 0 0.2 0.4 0.6 0.8

(b)-1.48

-1.475

-1.47

-1.465

-1.46

-1.455

0.725 0.73 0.735 0.74 0.745

Figure 26. (a) Parameter space overview for system (8.1). The bifurcation curves S0 (saddle-node of

equilibria), H (Hopf), and ha/b0 (two E-homoclinic orbits) emanate from the codimension-two saddle-node

Hopf bifurcation SNH. The curve of saddle-node bifurcations of limit-cycles Sl emanates from the degenerateHopf point DH on the curve H, and the two curves of E-homoclinics, ha

1 and hb1, emanate from noncentral

saddle-node homoclinic bifurcation points NS on S0. Panel (b) shows a magnification of the part of parameterspace in which the curve hb

1 terminates.

E is real and at least one EP1t-point at which there is a complex conjugate pair of leadingeigenvalues at E.

8.1. A model from the study of semiconductor lasers. Motivated by studies of semicon-ductor lasers with optical reinjection, the following system of equations was proposed in [20]as a model of a saddle-node Hopf bifurcation with global reinjection:

x = ν1x − y + x sin ϕ − (x2 + y2)x + 0.01(2 cos ϕ + ν2)2,

y = ν1y + x + y sin ϕ − (x2 + y2)y + 0.01π(2 cos ϕ + ν2)2,(8.1)

ϕ = ν2 − (x2 + y2) + 2 cos ϕ.

The bifurcation parameters for this vector field are ν1 and ν2. The system reduces to astandard normal form for a saddle-node Hopf (SNH) bifurcation if a Taylor expansion isperformed up to third order at the points (x, y, ϕ) = (0, 0, 0) at (ν1, ν2) = (0,−2) and (0, 0, π)at (0, 2). However, the variable ϕ is periodic, which causes reinjection and the existence ofnew orbits, which wind around a cylinder, that would not be present if ϕ were not periodic.

It was shown in [20] that there exists a curve of E-homoclinic orbits to an equilibrium,b, that emanates from a noncentral saddle-node homoclinic bifurcation point NS; this curveof E-homoclinics is labeled hb

1 in Figure 26. Panel (b) of that figure shows how hb1 snakes

toward a segment connecting two EP1t-points. The eigenvalues of the equilibrium b at thetwo EP1t-points are approximately −1.360 and 1.418 ± i at (ν1, ν2) = (0.737858,−1.46667)and −1.370 and 1.427 ± i at (ν1, ν2) = (0.742520,−1.45729). The Floquet multipliers of theperiodic orbit Γ are 1, 0.4994, and 0.526× 103 at the left-hand EP1t-point and 1, 0.4980, and0.636×103 at the right-hand point. Thus both EP1t-points conform to Case III of our analysis.Note that the complex eigenvalues of the equilibrium are reversed in sign with respect to theassumptions made in our analysis, and so we need to reverse time to compare with the theory.


This system was investigated further in [21], and additional bifurcation curves were com-puted, including cb, the locus of a heteroclinic orbit connecting Γ to b, tb, the locus of hete-roclinic tangencies of W u(b) and W s(Γ), and tΓ, a curve of P -homoclinic tangencies. It wasshown in [21] that the curves tb form the approximate envelope of the E-homoclinic curvehb

1; the turning points of hb1 are all approximately on the curves tb and accumulate on tb

extremely quickly. Reference [21] also showed that the curve c∗b , which is the locus of a secondheteroclinic orbit connecting Γ to b, accumulates in a way similar to the E-homoclinic curve,snaking between two P -homoclinic tangency curves, tΓ, and accumulating on a segment ofthe curve cb.

In order to compare this model with the theoretical results obtained in sections 3.2, 6.2,and 7.1, we apply a linear transformation to parameter space. This is possible becausefrom [21] we know that both the EP1 segment connecting the two EP1t-points and the curvestb between which the E-homoclinic curve snakes are linear up to numerical accuracy. Thelinear(-affine) map we use has the form

(8.2)(

μ1

μ2

)=(

a bc d

)(ν1

ν2

)+(

fg

),

where the coefficients a, b, c, d, f , and g are determined by specifying the image of three points:the two EP1t-points are mapped to the μ2-axis (specifically, to (0, 0) and to approximately(0,−9.4×10−3)) and one of the (relatively high order) turning points of the homoclinic snakeis mapped to the μ1-axis (specifically, the turning point called q9 in Table 1, which lies veryclose to heteroclinic tangency tb, is mapped to the point (μ1, μ2) ≈ (5.5 × 10−6, 0)). Theserequirements result in approximate values of the constants a = −1.73, b = 0.859, c = −1.46,d = 1.73, f = 2.54, g = 3.61. Applying this mapping yields Figure 27, which shows theE-homoclinic in the new coordinate frame; panel (a) shows the transformed snake alone, andpanel (b) shows the snake with some curves of folds of limit cycles superimposed.

In this new coordinate frame it is straightforward to extract the horizontal scaling law forthe E-homoclinic snake. The theory for this case (Case III) predicts that the scale factor bywhich the μ1-coordinate of successive turning points reduces is equal to the stable Floquetmultiplier, i.e., 0.4994 for the minima and 0.4980 for the maxima. As summarized in Table 1,the horizontal scaling of the homoclinic snake matches this prediction up to almost fourdecimal places. Due to more accurate calculations, this gives an improvement in numericalaccuracy of almost two decimal places compared with the results of [20]. It is not possibleto extract accurate estimates of the vertical scaling from this data: the vertical scale factoris predicted to be equal to the reciprocal of the unstable Floquet multiplier, i.e., more than200 times smaller than the horizontal scale factor, and appears to be dwarfed by higher ordereffects. Figures 28 and 29 show magnifications of the top and bottom parts of Figure 27(b),in particular showing the folds of limit cycles. Notice the striking similarity to Figure 20.

Finally, we show numerical evidence for the existence of P -homoclinic tangencies nearthe purported EP1t-points. Following the periodic orbit S8

l along the path in parameterspace indicated by the dashed blue line in Figures 27 yields the bifurcation diagram in Figure30(a). It can been seen that the periodic orbit emanates from an E-homoclinic bifurcationat ν2 ≈ −1.463, passes through a lot of folds, and eventually appears to approach some P -homoclinic tangencies (presumably one tangency corresponding to each accumulation point


Table 1Table of scalings for turning points of the E-homoclinic snake in system (8.1). Here, δi(μj) = |(μj(pi) −

μj(pi−1))/(μj(pi−1) − μj(pi−2))|, Δi(μj) = |(μj(qi) − μj(qi−1))/(μj(qi−1) − μj(qi−2))| for j = 1, 2. For theminima, the computed values of the Floquet multipliers predict a horizontal scale factor of 0.4994 (cf. δi(μ1)).For the maxima, a horizontal scale factor of 0.4980 is predicted (cf. Δi(μ1)). The vertical scale factors cannotbe extracted accurately from this data.

Minima on curve hb1

Point pi μ1(pi) μ2(pi) δi(μ1) δi(μ2)

p1 2.3250947336 × 10−3 −9.2964538119 × 10−3

p2 1.1113889215 × 10−3 −9.3468163580 × 10−3

p3 5.4190197325 × 10−4 −9.3626660243 × 10−3 0.4692 0.3147p4 2.6706866959 × 10−4 −9.3686543262 × 10−3 0.4826 0.3778p5 1.3239903424 × 10−4 −9.3712095355 × 10−3 0.4900 0.4267p6 6.5850910620 × 10−5 −9.3723821246 × 10−3 0.4942 0.4589p7 3.2810446805 × 10−5 −9.3729424933 × 10−3 0.4965 0.4779p8 1.6363181270 × 10−5 −9.3732161068 × 10−3 0.4978 0.4883p9 8.1640088701 × 10−6 −9.3733511728 × 10−3 0.4985 0.4946

Maxima on curve hb1

Point qi μ1(qi) μ2(qi) Δi(μ1) Δi(μ2)

q1 1.5441614595 × 10−3 −1.1008437384 × 10−5

q2 7.5158021016 × 10−4 −2.4937897816 × 10−6

q3 3.6926819386 × 10−4 −5.8516346851 × 10−7 0.4824 0.2242q4 1.8246714325 × 10−4 −1.3982899258 × 10−7 0.4886 0.2333q5 9.0466481392 × 10−5 −3.3618422283 × 10−8 0.4925 0.2385q6 4.4940168388 × 10−5 −8.0767046960 × 10−9 0.4948 0.2405q7 2.2349153804 × 10−5 −1.9906733630 × 10−9 0.4962 0.2383q8 1.1121307124 × 10−5 −5.7585064323 × 10−10 0.4970 0.2325q9 5.5360633840 × 10−6 0.0000000000 0.4974 0.4070

-0.01

-0.008

-0.006

-0.004

-0.002

0

0 0.01 0.02 0.03 0.04 0.05 0.06

hb1

(a)

μ1

μ2(b)

hb1S8

l

S6l

μ1

μ2

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0

Figure 27. Figure 26(b) after a linear mapping of the parameter space as defined by (8.2): (a) Thetransformed homoclinic snake; (b) the snake with curves of folds of limit cycles (labeled Sn

l ) superimposed.Here n denotes the number of loops the associated periodic orbit makes in the vicinity of the orbit Γ. Thedashed blue line in panel (b) indicates the path in parameter space used in constructing Figure 30. The largedots on this line show the accumulation points for the folds of the periodic orbit shown in Figure 30(a).


-0.0002

0

0.0002

0.0004

0.0006

0.0008

0.001

-0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007

(a)

S20l

hb1

S8l S6

l

μ1

μ2

-5e-06

0

5e-06

1e-05

1.5e-05

0 0.001

(b)

Figure 28. (a) Blow-up of Figure 27(b) around μ2 = 0. (b) Blow-up of the boxed area around (0, 0) inpanel (a).

(a)

S6lS8

l

S20l

�

hb1

μ1

μ2

-0.00936

-0.00934

-0.00932

-0.0093

-0.00928

0 0.0005 0.001 0.0015 0.002 0.0025

-0.00936

-0.00935

-0.00934

0.00052 0.00056 0.0006 0.00064

(b)

(c)

Figure 29. (a) Blow-up of Figure 27(b) around μ2 = −0.0093. (b) Blow-up of the boxed area around(0.0006,−0.00935) in panel (a). (c) Further blow-up of the boxed area in panel (b). The dashed blue line inpanel (a) indicates the path in parameter space used in constructing Figure 30.

of folds of the periodic orbit). Figure 30(b) shows phase space pictures at one of the foldbifurcations of lower order. Here, we use a more convenient representation of global periodicand homoclinic orbits in R

3, so that it is immediately clear how they close up, as in [20]. Thisrepresentation was achieved by mapping the ϕ-axis onto a circle with sufficiently large radius,R = 2, using the transformation

u = (R + x) cos ϕ,

v = (R + x) sin ϕ,

w = y.

Reference [4] shows a picture similar to Figure 30 but for the FitzHugh–Nagumo equations.


50

60

70

80

90

100

-1.464 -1.462 -1.46 -1.458ν2

T

(a)

-2 -1 0 1 2 3-2-1

0 1

2 3

-1

-0.5

0

0.5

1

u

w

v

(b)

-1

-0.5

0

0.5

1

0 20 40

w

T

Figure 30. (a) Transition of the periodic orbit S8l between an E-homoclinic bifurcation and a pair of P -

homoclinic tangencies along the path through parameter space indicated by the dashed blue line in Figure 27.For this path, ν1 ≈ 0.73961. (b) The periodic orbit for the parameter values at the large dot near the label (a)in panel (a), in (u, v, w) space (top), and for w versus time (bottom).

8.2. A model of intracellular calcium dynamics. The equations we consider in this sub-section are traveling wave equations derived from a model for calcium wave propagation inpancreatic acinar cells. The model was originally developed in [34] and [35] and further studiedin [33]:

c = d,

Dcd = sd − J,

sce = γ(Jserca − (kfPIPR + ν1Pryr + Jer)(ce − c)),

sR = ϕ−2O − ϕ2pR + (k−1 + l−2)I1 − ϕ1R,

sO = ϕ2pR − (ϕ−2 + ϕ4 + ϕ3)O + ϕ−4A + k−3S,(8.3)

sA = ϕ4O − (ϕ−4 + ϕ5)A + (k−1 + l−2)I2,

sI1 = ϕ1R − (k−1 + l−2)I1,

sI2 = ϕ5A − (k−1 + l−2)I2,

sw = kcm(w∞ − w)/w∞.

Here c denotes the concentration of calcium ions in the cytoplasm, ce gives the concentrationof calcium ions in the endoplasmic reticulum, and the variables R, O, A, I1, and I2 denote thefraction of receptors in various states. The bifurcation parameters used are s (the wave speed)and p (the concentration of inositol trisphosphate in the cytoplasm). The definitions of thequantities J , Jserca, etc. and the values of all constants used in the numerical computationsusing these equations are given in the appendix.

As described in [4], a branch of E-homoclinic bifurcations is seen to fold many times (ina “homoclinic snake”) for s ≈ 6.7 and p varying between about 1.88 and 2.63. Figure 31shows various parts of the bifurcation diagram and some representative phase portraits for


74

74.5

75

75.5

76

1.6 1.8 2 2.2 2.4 2.6

e

�d�c��b

68

70

72

74

0 0.05 0.1 0.15

(a) (b)

L2-norm

p c

ce

68

70

72

74

0 0.05 0.1 0.15 68

70

72

74

0 0.05 0.1 0.15

(c) (d)

ce ce

cc

f

(e) (f)

L2-norm L2-norm

pp

75.3

75.4

75.5

1.88 1.9 1.92 1.94 1.96

75.3

75.4

75.5

1.88 1.9 1.92 1.94 1.96

75.32

75.34

75.36

1.93 1.94 1.95

Figure 31. Snaking diagram of E-homoclinic orbits in the nine-dimensional calcium dynamics model,equations (8.3). Panel (a) shows the location of all the turning points (black points) of E-homoclinic orbits inthe parameter-versus-L2 -norm plane. The homoclinic orbit and associated periodic orbit at turning points b, c,and d, near the right edge of the snaking envelope, are depicted in the corresponding panels (b)–(d). Panels (e)and (f) show two successive magnifications of panel (a) near a complex sequence of internal folds.

this phenomenon. A feature of this bifurcation diagram that we highlight is that turningpoints of the snake accumulate on six different values of p, which suggests that there are infact six cyclically connected EP1t-points and associated turning points as in the case discussedin section 7.1 and illustrated in Figure 24.

AUTO is not reliable for finding Floquet multipliers in such a stiff system, and henceit is hard to compare the theoretical predictions for the rate of convergence of the folds ofE-homoclinics with what is seen in the numerical bifurcation set. Furthermore, our analysisis restricted to R

3 and so may not apply to this example. However, the phenomenon seenin this system is qualitatively the same as predicted by our theory. There is extremely fastconvergence of the turning points of the homoclinic snake (the different turning points on theE-homoclinic curve are numerically indistinguishable in the (p, s)-parameter space), which isconsistent with our theory in the case that the system is stiff. The lack of reliable estimatesof the Floquet multipliers prevents us from determining which case of the theory might berelevant to this system (i.e., positive Floquet multipliers or negative Floquet multipliers). Theeigenvalues of the equilibrium can, however, be determined. At the left edge of the snakingenvelope the eigenvalues are computed to be −4.32, −3.82, −2.01, −0.131, −0.031 ± 0.138i,


−0.0144, −0.00673, and 0.4352564, while the eigenvalues at the right edge of the envelope are−5.24, −4.48, −2.13, −0.137, −0.0151, −0.0922, −0.000577± 0.1485265i, and 0.3613053. Wenote that the two leading eigenvalues for the left edge of the snaking envelope are real, andthose for the right edge are complex, but this difference does not affect the analysis of theE-homoclinic bifurcations, as discussed in section 5.

9. Conclusion. This paper has (partially) unfolded the dynamics occurring near an EP1t-point, i.e., near a codimension-two heteroclinic cycle connecting a hyperbolic equilibrium and ahyperbolic periodic orbit in the case that the connection from the equilibrium to the periodicorbit is of codimension one and the connection from the periodic orbit to the equilibriumoccurs at a tangency of the unstable manifold of the periodic orbit and the stable manifoldof the equilibrium. We were initially motivated to perform this study as part of a widerproject which aimed to categorize some of the ways in which branches of homoclinic orbits ofequilibria can terminate near Hopf bifurcations in excitable systems [4]; EP1t-points turn outto be relevant in this context.

We have performed a geometric analysis that has enabled us to locate homoclinic bifur-cations of the equilibrium, homoclinic tangencies of the periodic orbit, and folds of periodicorbits, all in a neighborhood of an EP1t-point, and to deduce scaling laws for these bifurca-tions. Using simple geometric assumptions about the way in which two or more EP1t-pointsmight be globally connected (in parameter space), we have shown why wide snaking curves ofhomoclinic bifurcations of the equilibrium are seen in some systems with EP1t-points. Ourmain results are summarized in section 2 and illustrated with numerical examples in section 8.

Our analysis has been via standard methods, i.e., the construction and analysis of geo-metric return maps valid in a neighborhood of an EP1t-point. We have not given completeproofs of the validity of our model as an approximation to the dynamics near the heterocliniccycle, choosing to focus instead on providing a simple picture of the behavior of the maincodimension-one bifurcations that occur, but we believe that a rigorous proof is possible.

Our analysis has been restricted to R3, the lowest possible phase space dimension in which

an EP1t-point can occur. It seems likely that results such as the homoclinic center manifoldtheorem and techniques such as Lin’s method could be used to extend the results to higherdimensions, but we have not attempted to do so. However, we note that one of the numericalexamples we considered was nine-dimensional, and the main results of our analysis seemedto apply to that example, lending credence to our view that the results in this paper will bevalid beyond R

3.We have performed only a partial unfolding of an EP1t-point. Aspects of the dynamics

that remain to be investigated include the existence of period-doubling bifurcations of periodicorbits, the presence of chaotic dynamics, and bifurcations involving multipulse homoclinicorbits (passing more than once near the equilibrium). There are also other related cases thatcould be considered. For instance, analogous calculations could cast light upon the dynamicsseen near codimension-two EP2-cycles such as those described in [28]. Lastly, in the contextof excitable systems, from which our interest in EP1t-points arose, the homoclinic orbits ofinterest in our study actually correspond to isolated traveling pulses in the underlying partialdifferential equation models; it would be of interest to investigate any implications for pulsestability arising from the dynamics we have uncovered. All of these phenomena remain thesubject of future work.


Appendix. Quantities and constants for the calcium model equations. In numericalcomputations using (8.3), the following quantities and constants were used:

J = (kf PIPR + ν1Pryr + Jer)(ce − c) − Jserca − Jmito + δ(Jin − Jpm),

Ka = (kam/kap)14 , Kb = (kbm/kbp)

13 , Kc = kcm/kcp,

l−2 = l2k−1/(k1L1), l−6 = k−4l6/(k4L5), l−4 = k−2l4/(k2L3),

w∞ =1 + (Ka/c)4 + (c/Kb)3

1 + 1/Kc + (Ka/c)4 + (c/Kb)3, Jmito = Vmito

c3

1 + c2,

Pryr = w1 + (c/Kb)3

1 + (Ka/c)4 + (c/Kb)3, Jin = Jinbase + 0.05p,

S = 1 − R − O − A − I1 − I2, PIPR(O,A) = (O/10 + 9A/10)4,

ϕ1(c) =(k1L1 + l2)c

L1 + c(1 + L1/L3), ϕ2(c) =

k2L3 + l4c

L3 + c(1 + L3/L1),

ϕ−2(c) =k−2 + l−4c

1 + c/L5, ϕ3(c) =

k3L5

L5 + c,

ϕ4(c) =(k4L5 + l6)c

L5 + c, ϕ−4(c) =

L1(k−4 + l−6)L1 + c

,

ϕ5(c) =(k1L1 + l2)c

L1 + c,

Φ2 = ϕ−2/ϕ2, Φ4 = ϕ−4/ϕ4,

A = pR/(p + Φ2Φ4 + pΦ4), JIPR = kfA4,

Jpm(c) =Vpmc2

K2pm + c2

, Jserca(c, ce) =Vsercac

ce(Kserca + c).

Jinbase Jer γ Vserca Kserca Vpm Kpm δ Dc ν1 Vmito

0.2 0.002 5.405 120.0 0.18 28.0 0.425 0.1 20 0.04 0

kf kap kam kbp kbm kcp kcm l2 l6 L1 L5

0.32 1500 28.8 1500 385.9 1.75 0.1 1.7 4707 0.12 54.7

l4 L3 k1 k−1 k2 k−2 k3 k−3 k4 k−4

1.7 0.025 0.64 0.04 37.4 1.4 0.11 29.8 4 0.54

Acknowledgments. The authors acknowledge helpful conversations with Bernd Kraus-kopf, David Simpson, James Sneyd, and Arnd Scheel.

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unfolding a tangent equilibrium-to-periodic heteroclinic...

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